This manual contains instructions to get you started using Stella4D.
It covers most features, and after reading this document you should be able to
find your way around most of the interface.
A description of how to use various features will be found here, but for a more
detailed explanation of what is possible and what the features can be used for,
please see my paper
"Stella: Polyhedron Navigator".

Note: it becomes difficult or convoluted to describe some features in a
generic way that covers both the 3D and 4D cases, and would confuse those
people more familiar with 3D. For this reason, I will explain most features
using a language specific to the 3D case, and save 4D specifics for the
4D
section. So often, when I refer to a "polyhedron", the same will also hold for
a "polychoron" (a 4D polytope). Similarly in 4D, "plane" may mean
"hyperplane", "sphere" may mean "hypersphere", and "face" may mean "cell".

The program includes all the uniform polyhedra (which includes the Platonic,
Archimedean and Kepler-Poinsot solids and more), the Johnson solids, some
Stewart toroids, compounds, and other polyhedra.
In 4D, it also includes all 1849 known uniform 4D polytopes, plus
duoprisms, antiduoprisms, a selection of scaliforms and more.

The default window layout has two views side-by-side, showing the
base polyhedron on the left and its net (or one of its nets) on the
right. The default base model is the icosahedron.

Use the Right and Left arrow keys to change to the
next/previous model in the built-in list (or the menu items
"Poly→Next Polyhedron" and
"Poly→Previous Polyhedron", or the green left and right arrow
buttons on the main toolbar or at the top of some views). Note: if
you have loaded a .stel file, then the arrow keys move forward and
backward through files in the same folder instead. Furthermore, the arrow
keys, unlike the green arrow buttons and menu items, are context
sensitive. That is, they behave differently depending on what type of view
is currently active. If the current view has yellow arrow buttons at the
top, then they performs the function of those instead (e.g. changing to the
previous/next net if the unfolded net view is active). See more about yellow
arrow buttons later. You can also use Ctrl+Left and
Ctrl+Right to go to the previous/next model, regardless of the
selected view (they behave exactly like the green arrow buttons).

The built-in list is divided into a hierarchy of categories. You will see
the current category and model name displayed in the main toolbar. Clicking on
either of these names opens a list of other categories or models to choose
from, providing a more direct way to select a model.

Archimedean: The 13 convex uniform (but not regular)
polyhedra. There's no category for their duals,
the Catalan solids, because the dual of any polyhedron is
instantly available in Stella4D already. So access them via this
list.

Tetrahedral Symmetry: The one and only nonconvex uniform
polyhedron with tetrahedral symmetry.

Icosahedral Symmetry: The 32 nonconvex uniform (but not
regular) polyhedra with icosahedral symmetry, not including snubs.

Nonconvex Snubs: The remaining 10 uniform polyhedra, all
snub models with icosahedral symmetry, plus Skilling's figure, which is
also uniform, but not officially a true polyhedron since four faces
meet at some edges. Snub models have some faces that do not
share any rotational symmetry with the model.

Degenerates: Uniform models with four faces meeting at each
edge. If faces were connected differently, these models could also be
seen as degenerating into a compound of two other uniform polyhedra.
Many of these just look like some other simpler model, but there are other
faces hidden inside. Try hiding some outer faces to see inside, or
explode the faces apart with
Ctrl+Shift+Left-drag. You'll also notice differences in the way
they morph between duals.

Johnson Solids: The 92 remaining convex regular-faced polyhedra.

Near Misses: Some models that are "almost" Johnson solids. In each
case some of the faces are not quite regular.

Stewart Toroids: Regular-faced polyhedra with genus greater than
zero, i.e. with holes. Most are not self-intersecting. A few regular-faced
polyhedra with genus zero are included. Stewart used these as building blocks
for some of his toroidal models. See Stewart's book
Adventures Among the Toroids for more information.

Stella Library: This category contains extra models that are
not actually built into the program, but rather come from an additional library
of models provided with Stella4D. These reside in a folder called
"StellaLib" under the folder where Stella4D is installed.
You can add your own .stel files to the folder and they will also
appear in this category. (See Polyhedron Library).

4D Library: Similar to above, but this category contains 4D models,
which reside in a folder called "Stella4DLib" under the folder where
Stella4D is installed.
(See Loading 4D Polytopes).

Geomag Library: This category contains fun models which can be made
using Geomag's construction kit. These reside in a
folder called "GeomagLib" under the folder where Stella4D is
installed.
You can add your own .stel files to the folder and they will also
appear in this category.

Another way to find a polyhedron is to use
"File→Polyhedron List...", hit Ctrl+N, or click the
matching button on the main toolbar. This opens a window listing all the
polyhedra provided. A list of categories on the left, and a list of polyhedra
from the current category on the right. The list of polyhedra has several
columns, including the primary symbol, name, and the number of faces, edges,
and vertices. The list may be sorted according to any column by clicking on
the header of that column. Click a second time to reverse the order. To
return to the original order, click on the header of the first column.

Select a category on the left, then select a polyhedron on the right.
Information about the selection appears at the bottom, including other common
names, dual names, Wythoff symbol, Wenninger number and page number in
Wenninger's books Polyhedron Models and Dual Models where the
model can be found. Click OK or hit Enter to open the selected
polyhedron.

Rather than looking through the list, you can enter a symbol or name
directly in the Search field at the top. Symbols for uniform polyhedra
may be any of the following:

Vertex descriptions (separated by either commas or dots). E.g.
"4.4.4" (cube)
or "(3.3.3.3.3)/2" (great icosahedron).
For the infinite series of prisms and antiprisms you may enter "23.4.4" or
"13.3.3.3" etc. You may even create compounds of prisms or antiprisms by using
fractions with a common factor on the top and bottom, e.g. "6/2.4.4" or
"8/3.2.2.2", and degenerate antiprism compounds where the top of the fraction
is double the bottom, e.g. "8/4.3.3.3" (here the top and base of the antiprism
have degenerated into a single edge).

Index in Magnus Wenninger's book Polyhedron Models, entered with a
"W" prefix. E.g. "W3" (cube)
or "W41" (great icosahedron).
You may even enter numbers for stellations, such as "W54" (a
stellation of the icosidodecahedron). In this case you'll be prompted to
switch to a stellation view if you don't already have one open.

Jonathan Bowers's index for the uniform 4D polytopes, entered without a
prefix. E.g. "12" or "1846".

Other models use different notations. For example, enter "J4" for the
square cupola, one of the Johnson solids. This provides a very quick way to
jump to a particular Johnson solid if you know its number.

You may also enter the name of a polyhedron. It can be either the full
name (including known alternative names and dual names), a substring in the
name, or an abbreviation for the name using either the first letter from each
word, or the first letter from each part of each word. Some examples:

"RTC". Abbreviation for rhombic triacontahedron.
Since this is the dual of a built-in model, rather than being a built-in model
itself, the icosidodecahedron is the model that will be loaded, but you can see
the RTC in the dual view, and will be prompted to switch to a dual view if you
don't have one open already.

Part of a file name from the library will also work. E.g. "cubes 5" for
the 5-cube compound or "Bruckner 26,1".

Jonathan Bowers's abbreviated names for 3D or 4D polytopes are mostly
accepted too. E.g. "gaghi" or "ondip". Occasionally there are name clashes.
If you enter the name of a 4D polytope and it finds the wrong model, try
putting a dash at the start, e.g. "-pen" or "-gishi".

As you type into the search field, you'll notice it finding the best match as
you go. Hit Enter to load a model as soon as it finds the one you're
after.

Below the search field is a choice between Find one and Find
all. So far we've been using the default, Find one, which means the
first match is found. Find all finds all matches, and lists them in the
special Search Results category at the top of the category list. Select
an item from the list and open it. Now you can use the Left and
Right arrow keys (or green arrow buttons) to step through the search
results.

To the right of the big Search field are some smaller fields where
you can enter
cell (for 4D polytopes),
face, edge and vertex counts. If you enter a number, this also becomes part of
the search. For example, enter "12" in faces, and only polyhedra with 12 faces
will be found. A range may be entered, such as "12-24", to find all polyhedra
with between 12 and 24 faces. You may enter values into more than one field to
further narrow the search. Comprehensive tooltips help explain how to use each
field, so hover the mouse them to find out more.

With the Search Results category and Find all selected,
another button appears labelled ←Locate search result. Click this
to locate the selected polyhedron in its original category.

Every polyhedron has a dual, which won't be explained fully here, but
you may think of it as the model's opposite. It has the number of faces
and vertices swapped with respect to the original model, and the same number of
edges. The dual of the dual brings us back to the original model again.

Hit "d" to switch between the base polyhedron and its dual. The
net will also change.

The closer a facial plane passes to the centre of a model, the further away
from the centre the corresponding dual's vertex will be. If the facial plane
passes right through the model's centre, then the dual vertex will be
infinitely far away (in a direction perpendicular to the face). Although
Stella4D doesn't allow base polyhedra to have infinite parts, their
duals can. In the dual view you can see these as faces that extend out towards
infinity, but stop after a fixed distance. You can control this distance
interactively with Ctrl+Left-drag. Try it, moving the mouse
left and right, to show more or less of the infinite dual faces. This will
only work if the dual model does indeed have infinite parts of course.

At any point you may save a model you like to Stella4D's native format,
the .stel file. Everything about your scene is saved here, including
your current screen layout and views, the model itself,
the current stellation of that model, any
facets created in Faceting Mode,
various settings, and the state of mouse inertia. The
latter means that if your model is spinning, morphing or folding etc. when you
save it, then it will be spinning, morphing or folding again when you reload
it.

Use "File→Save" or "File→Save As" to save your
model, and "File→Open" to re-open it later. In the file browser
that appears, select an existing .stel file to see some information
about it in the preview area at the bottom of the browser. You will see the
number of faces, edges and vertices, and the first line of the comment field
from the Info Window if one has been set.

Tip: Any .stel files saved from Stella4D before version 5.0 will
need to be loaded and re-saved in order to support this information preview in
the browser.

If you particularly like a model, you may make it load immediately when
Stella4D starts up by using "File→Save Default Scene". Next
time you start Stella4D, this model will load automatically. You may even
use this just to set up your view layout the way you like
it.

The mouse does many different things, depending on what mode you're in,
which view you're in, whether you're holding down the left/right/both mouse
buttons, and whether you're holding down Shift/Ctrl/Neither/Both/Space. Watch
for the tips in the bottom right hand corner, which show you what the mouse
buttons do in the current situation. The tips change when you hold down
Shift/Ctrl/Neither/Both/Space. They may also change when you move the pointer
from one view to another. Even I forget what the mouse can do in some modes,
so these on-screen tips help a lot!

Sometimes just clicking is required (e.g. Shift+Left-click
to select a face). Sometimes dragging in 2D is required (e.g.
Left-Drag to tumble the polyhedron). And sometimes dragging in 1D is
required (e.g. Right-Drag to zoom in/out).

All basic navigation is done with the mouse and no need to touch the
keyboard, except for some less common movements. These mouse controls
generally continue to work in different modes too.

In a 3D view:

Left-drag: tumble/rotate

Right-drag: zoom in or out

Left+Right-drag or Middle-drag: twist
(rotate around the viewing axis). Left+Right-drag means to hold down both the
left and right mouse buttons while moving the mouse. Any time
Left+Right can be used in Stella4D, you may use the
Middle button instead, if your mouse has one.

Tumbling and twisting both have mouse inertia, so
you can release the mouse buttons while dragging and the model will continue to
tumble or twist. To stop it, perform the same action again, but clicking
instead of dragging the mouse (e.g. do Left-click to stop tumbling).
You may also hit Esc to stop any mouse inertia.

Similarly in a 2D view:

Left-drag: pan sideways

Right-drag: zoom in or out

Left+Right-drag: twist (rotate)

With 2D views, zooming zooms in on the point where the mouse was when you first
clicked the right button, so you can zoom in on a specific point. As with 3D
views, twisting has mouse inertia.

A couple of more advanced navigation controls are available in 3D views.
Hold down the Space bar and you'll notice the mouse-tips in the bottom
right corner change:

Space+Left+Right-drag: pivot around an axis. Select a
face, edge, vertex, or rotational symmetry axis (see
Symmetries), and this operation will pivot around an
axis through that selected item. You may deselect the item and the axis will
be remembered (until another item is selected).

If you find that you want to pivot around a selected axis more often than
twisting, select "Options→Right-drag to Pivot Around Selected
Item" (or matching toolbar button). This swaps the operations of
Right-drag and Space+Right-drag.

You may also switch each view between perspective and orthogonal projections
by ticking/unticking "View→Orthogonal View" (keyboard shortcut:
O). Perspective views are what we see in the real world, where things
closer to the camera appear to be larger. Orthogonal views are like
architectural plans, where distance from the camera does not affect the
apparent size of objects.

Finally, you can use items on the "View→Camera" submenu to
store and recall camera positions. The field-of-view and
perspective/orthogonal setting are also stored.

Ctrl+Shift+Right-click: Unexplode faces, bringing them back
to where they started.

This can be useful for examining the internal structure of faces that would
otherwise be partly or entirely hidden.

Exploding has inertia, so if you release the mouse
button while still dragging, the exploding will continue at the current rate.
It will remember how far apart the faces are and use this as one extreme in the
explosion. So faces will explode out to this point, then turn around and
implode again. The other extreme, at the imploded end, is determined by the
following menu item:

Options→Allow Explosion Inertia to Implode. When ticked, faces
will implode back through each other and out the other side, to the same
distance as the extreme in the other direction. When not ticked, faces will
only implode back to their original positions before changing direction and
exploding again.

Exploding faces is available in most 3D views, but not the Unfolding Net view.

In 4D, the behaviour is a little different. Instead of exploding faces
apart, cells are grown or shrunk in-place.

Stop all mouse inertia. This is very handy when
things are spinning and morphing and folding and exploding and you can't
remember what combination of Shift, Ctrl, Space, and
Left and/or Right mouse buttons you used to start all that
happening!

In the default mouse selection mode(see below),
faces, edges and vertices may be selected with the mouse.

Shift+Left-click or double Left-click:
Select a face, edge, vertex, or symmetry axis. The selected item is
highlighted in white and will partly show through other faces (try rotating the
model so that the item is on the other side). Only one item may be selected at
a time. Many operations work on the current item, which is either the
selected item, or the most recently selected face if no item is selected. If a
face has not yet been selected, then the first face is the current face.

Ctrl+Left-click: Select only faces (not an edge or vertex).

Ctrl+Right-click: Select only edges.

Shift+Right-click or double Right-click:
Select only vertices. A selected vertex is highlighted with a white dot, and
half of each surrounding edge is also highlighted. This also selects the
corresponding face of the dual, which will also become
highlighted if you have a dual view open.
Selecting a vertex is useful in a few situations, such as when you have a
vertex figure view open and want to choose which vertex figure to look
at.

There are five toolbars arranged in three rows, with two in the first and last
rows by default. They may be dragged and docked to different sides of the
window, or dragged away into a separate window. Their positions will be
remembered between sessions. The toolbars are:

Main toolbar: the top toolbar. Provides buttons for file
opening/saving, selecting polyhedra from the built-in list, and printing.

Tour toolbar: to the right of the main toolbar. Contains buttons
for loading, saving and creating tours, which are like animated
slideshows of polyhedra. See Tours.

Options toolbar: below the main toolbar. Provides buttons for
various options and operations.

Mode toolbar: left-hand toolbar under the options toolbar.
Contains buttons for changing the current mouse-selection mode,
which affects what the mouse does when holding down Shift and/or
Ctrl and clicking a mouse button. Normal navigation with the mouse
remains unchanged. Exactly one of these buttons will be pushed in at any time,
indicating the current mode. A helpful message appears when entering any mode
other than the default mode, and don't forget to keep an eye on the mouse tips
in the bottom right corner of the main window to see what the mouse does. You
can hit Esc to return to the default mode. See
Mouse Selection Modes for more information about modes.

View toolbar: right-hand toolbar under the options toolbar.
Contains buttons that change the current view to a different type. Exactly
one of the buttons on the view toolbar will be pushed in at any one time,
indicating the current view type. Try clicking through the different buttons
to see all the different kinds of view.

The buttons all have tool-tips, so if you place the mouse over a button
and don't move for a moment, a small description of the button appears.

In addition, there are further buttons in the top right corner of each view.
The buttons that appear depend on the type of view. Yellow left or right
arrow buttons are for changing to the previous or next item that this view type
can display. For example, in the 2D net view, these buttons cycle through the
various nets required. If the current view has yellow arrow buttons, then you
can use the left and right arrow keys on the keyboard to perform the same
function.
Occasionally you will also see tick and cross buttons, which are used to accept
or reject something. For example, these buttons appear in facet-creation mode,
and can be used to accept or cancel a partially completed facet.

The Info window is a special window for displaying information about
the current polyhedron. There are three ways to open or close it: via the
"View→Model Info" menu item; via the equivalent button on the
options toolbar; or by simply hitting "i" on the keyboard.

Information includes number of faces, vertices and edges, number of edges
that must be cut/folded/glued to make the model, alternative names for the
model (if any) and lots of other info. It is presented in a tree structure,
where collections of similar data are grouped together. A small "+" or "-"
sign beside each item may be clicked with the mouse, allowing each section
to be expanded to show all the data in that group, or collapsed
to hide the data. Which groups are expanded or collapsed is remembered between
sessions, so the data presented will always be the data of most interest to
you.

You can Right-click on information that may be edited to edit that
item. Most items can't be edited, but ones that can include the model's name,
the dual's name, comments about the model, internet link associated with the
model, radius, surface area and volume.

Expand the face, edge or vertex counts to see a list of face, edge or vertex
types. You can Left-click on a type to select a face, edge or vertex
of that type in the main view. Similarly, when a face, edge or vertex is
selected another way, the matching entry in the Info window will be
highlighted.

The Info window also shows how many parts belong to this compound (or "1" if
not a compound). Expand this item to see a list of the polyhedra in this
compound. Left-click on one to select a face belonging to that part.

This window starts off docked to the right hand side of the main window, but
like the toolbars, it may be dragged to dock elsewhere in the main window, or
dragged away into a free-floating window of its own. The position is
remembered between sessions.

By default there are two views, one showing the base polyhedron, and one
showing its net. You may choose a different layout with
"View→Choose Layout", or more conveniently by using
Ctrl+1 to Ctrl+6, depending on how many views you want. For
example Ctrl+4 will give you a four-view layout. Repeatedly hitting
Ctrl+4 will cycle through all the different four-view layouts
available.

Once you have the layout you want you can choose what kind of views are
shown by selecting each view in turn (by clicking in them or on their title
bars) and hitting one of the buttons on the view toolbar (or choosing from the
bottom section of the View menu).
Tip: To see which view is selected, look for the one with the
highlighted title bar. There's always exactly one selected view.

When you save a .stel file
(not available in the demo)
your layout and view types are also saved, and restored when the file is
opened another time. If you wish to open a file without changing the current
view layout, use "Options→Keep Layout when Opening Files". Once
ticked, you may continue to open further files without the layout being
affected.

There are eight types of view which show smooth morphing between a
polyhedron (or compound) and its dual.
Two of these are supported in 4D.
Select one of these view types from the view toolbar, or from the
"View→View Duals Morphing" submenu. Use Ctrl+Left-Drag
to morph between the base and dual in these views. There's also mouse
inertia on this function, so you can release the mouse button while
dragging and the morphing will continue on its own.
To create nets for a morphed model, you first need to make it the new base
model. Do this by clicking the left-and-down arrow button on the morph view's
title bar.

Note: the demo version will not allow morphing for certain models.

This morphing may not act perfectly between all pairs of models. Some
methods cause parts to get flatter and flatter until they disappear, which
produces a visual jump, but isn't really wrong.
Morphing between hemi-polyhedra and their infinite duals won't work
correctly, which probably isn't surprising!

Another type of view is the Unfolded Net view. Use PageUp and
PageDown to move through the list of nets required. Hit Ctrl+P to
print the net (or to print any other view type, but make sure you have the
appropriate view selected first!).
The demo version will only allow you to print nets for the five Platonic
solids, but it will still let you see a print-preview of nets for any model.
The Net view shows one unfolded net at a time, but when printing, all required
nets are included. Stella4D attempts to pack as many onto each page as
it can to minimize paper requirements.

Printing must always be done via a print-preview, to avoid any issues with
unexpected settings. A dialog box appears first full of options for printing
nets, but you can ignore most of these to start with and just click on
"Preview" (or hit Enter).

If you're not sure about how to put the nets together, then look for the
"Edge connection IDs" tick-box in the dialog box that appears when
you print nets. Ticking this will cause numbers to be displayed beside each
edge around the net. Each number will appear exactly twice among all the nets.
Find the matching numbers to see which edges should be glued together. This is
especially useful for models with asymmetric color schemes, where the nets may
go together in various ways, but only one has the correct colors.

When printing, if the sale of the model is too big for nets to fit on your
printer's paper size, an error is given, with an offer to scale the model down
just enough so that the nets fit. The message also shows the factor by which
the model would be scaled down.

The Folding Net view shows the nets in 3D, folding up into the final model and
unfolding again into separate flat nets. Use Ctrl+Left-drag to
interactively fold and unfold the nets. Mouse inertia applies here too,
so if you release the left mouse button while still moving the mouse, the
folding/unfolding will continue on its own at the current rate.

When unfolding, first the folded nets move apart from each other (if there's
more than one net), then they each unfold individually.
Ctrl+Right-click jumps to the point between these two stages, or just
folds the net half-way if there is only a single net.

The "Nets→Nets Shown in 3D View" submenu gives you control
over what is shown in the Folding Net view. Your options are:

Show All Nets

Show One Type of Net

Show One Net

If you want to try something more unusual, try creating the
convex hull of a partly folded net using
"Poly→Create Convex Hull" with the Folding net view selected!

You may want to print all nets onto white paper, perhaps using a color printer
to fill in the face colors or images, or you may want to
print nets of different colors separately in order to print directly onto
colored paper (my usual technique).

The "Nets→Net/Paper Color Mixing" submenu allows this.
It contains the following options:

Auto: Automatically choose between the two options below. Mixed
colors are used when the model is convex or has images on its faces. Otherwise
only a single color is permitted per net.

One Color per Net/Page: Only allow faces of a single color in each
net and on each page of paper when printing. Use this when printing onto
colored paper, as I usually do.

Allow Mixing of Colors: Faces of different colors may be combined
into single nets and printed onto the same pages. Use this if you want to use
white paper and a color printer to fill in the faces.

When mixing of colors is not permitted within nets, only nets of a single
color will be printed at any one time. For example, if there are red, yellow,
and blue parts, then you will need to do three print-outs to print all the
nets, one for each color. If you are currently viewing a yellow net in the
Unfolded Net View, then by default, printing will only print yellow nets, but
the dialog box has a drop-down list for choosing the paper color you want. For
the chosen color, all nets of that color will be printed, not just the one you
are viewing, and it will try to pack them together as best it can. The
print-preview lets you see how many pages are required, so you just put that
many pieces of red, yellow or blue paper in the printer tray (use manual feed
for thicker paper and to reduce curling on some printers).

In order to glue parts together when building a paper model, tabs are
usually left on some or all edges around each net. The following construction
methods are common:

No-tab method. Don't use tabs at all. Parts may be connected with
tape or similar. I don't recommend this method for paper models. Tape is ugly
and deteriorates over time.

Single-tab method. For each pair of edges to be glued together,
only one has a tab. It is glued directly under the part being attached.

Double-tab method. Tabs are left on all edges around each
net. When parts are attached, it is their tabs that are glued together,
forming a kind of ribbing inside the model.

Most people seem familiar with the single-tab method, and indeed it's how I
first started, but for the most part I recommend the double-tab method for the
following reasons:

Construction is easier. If using tweezers to squash the two tabs together
when gluing, it's generally easier to access them. With a single tab glued
under the connecting face, other parts of the model may get in the way.

I think the result generally looks better, more precise.

Gluing directly under a face that will be visible from outside can cause
problems. Sometimes the water in the glue can cause wrinkling. It also
matters more if you make a mess with the glue. Glue may get on the tweezers
or your fingers and be spread onto the face when squashed together with the
tab.

It's also generally easier to put the last piece in! With double-tabs,
the tabs fold at half the dihedral angle as compared with single-tabs, so they
have more spring in them.

The single-tab method can still be useful in the following cases:

When the angle between faces is very small, a single-tab can produce a
sharper result.

When there's not enough room in the net for two tabs (though in this case
you may want to think about rearranging the net).

You may mix and match methods within one model. In such cases I would start
by attaching the single-tabs, which are harder to glue once more of the model
takes form.

Stella4D supports all three methods, or a mix of them. Tabs are
shown in both the Unfolded and Folding Net views. Use the
"Nets→Tabs" submenu to state your preference. With an edge
selected, the preference applies to that edge only. With no edge selected, you
will be setting the default preference. It contains the
following options:

Use Default. Only available when an edge is selected. Tells the
edge to use the default setting. Other options below would tell it to override
the default.

No Tabs. No tabs will be created at all. If an edge is selected,
then don't put a tab on this edge or its partner, overriding the default.

Single-Tabs. A tab will be created on just one edge in each
edge-pair to be joined. If an edge is selected, then only use one tab between
this edge and its partner. Select this option a second time to flip which edge
has the tab.

Double-Tabs. Tabs will be created on all edges around each nets.
If an edge is selected, then a tab will be put on both this edge and its
partner, regardless of the global default.

Tab Size. Set how big you want your tabs. You can probably stick
with the default 0.5 cms.

Before printing out nets, you are going to want to decide how big to make your
model. This is what the items on the "Scale" menu are for. They let
you change the scale of the whole model by specifying the length of certain
features, such as edge length or radius. Note: the model does not get bigger
or smaller on the screen, but printed nets
or exported models
will reflect the change in scale. However, if you have tabs on your nets, then
you will see them change size. Visually, they scale in the opposite direction
to the change you make, because while the model might be physically bigger, it
remains the same size on the screen, so the tabs which have remained the same
physical size, are therefore shown to be smaller.

Tip: You may enter an equation rather than a
simple number if you wish.

Here's what the items on the Scale menu do:

Fit Nets to Pages
This adjusts the scale of the current model so that the largest net just fits
on a page. Use this if you want to make the model as big as possible, given
the nets you want to print out.

Base Polyhedron Radius...
Tells you the radius of the base polyhedron, and lets you change it.
Note: if you're making a stellation (see below), you probably want to use
"Scale→Stellation Radius" instead to set the radius of that
stellation.

Base Circumference...
Shows and allows you to edit the circumference of the base polyhedron's
circumsphere.

Edge Length...
Tells you the length of the selected edge, and allows you to change it. If no
edge is selected, it tells you the shortest and longest edge length in the base
or dual polyhedron (depending on selected view), and allows you to change the
longest edge length.

Base Polyhedron Inradius...
Tells you the inradius of the base polyhedron, if it has one. The inradius is
the radius of a sphere that just touches each face plane (that is, the face
planes are tangent to it). If there is not a single inradius, then it shows
you the distance of the selected face plane from the model's centre. You can
edit the value in either case to change the scale of the model.

Base Polyhedron Surface Area...
Tells you the surface area of the base polyhedron, and lets you change it.

Base Polyhedron Volume...
Tells you the volume of the base polyhedron, and lets you change it.

Dual Radius...
Tells you the radius of the dual polyhedron, and lets you change it.

Dual Inradius...
Tells you the inradius of the dual polyhedron, and lets you change it.

Dual Surface Area...
Tells you the surface area of the dual polyhedron, and lets you change it.

Dual Volume...
Tells you the volume of the dual polyhedron, and lets you change it.

Midradius / Reciprocation Radius...
Tells you the midradius of the polyhedron, and lets you change it. This value
is really the radius of the sphere used for reciprocation (the process used to
create the dual). For uniform polyhedra (e.g. Platonic and Archimedean solids)
it will be the midradius, i.e. the distance from the model's centre to lines
through each of its edges. For other models it may be an approximation, as
such models may not have a single midradius.

Stellation Radius...
Tells you the radius of the current stellation, and lets you change it.
See below for details about stellating.

Shortest Edge Length...
Tells you the length of the shortest required in the nets, and lets you change
it. This is useful because edges become too fiddly when they're too small. I
always check this before making a model.

Measured Distance...
Tells you the radius of the current measured distance when using
Measurement Mode, and lets you change it.

Stretch (Non-Uniform Scale)...
This item is different from the others above. Rather than just setting the
size of the model for which nets are printed, this item actually changes the
model, by stretching or squashing it in some direction. You may choose to
scale it along one of its symmetry axes, or perpendicular to the selected face,
along a selected edge, or through a selected vertex. You may scale by a given
factor, or set the resulting height of the model explicitly. You can even set
the resulting length of the currently measured distance in
Measurement Mode, which allows you, for example, to
set the resulting length of lateral edges of an antiprism.

Distance Units
Lets you choose what units to use for distances. Available units are:
millimetres, centimetres, metres, inches and feet.

Angle Units
Lets you choose what units to use for angles. Available units are:
degrees (decimal), degrees/minutes/seconds and radians. This affects how
angles are displayed when using "Nets→Show Edge Data→Dihedral
Angles", for example.

Tip: When entering a scale, or entering a real
number in any other part of the interface, you may enter an equation rather
than a simple number if you wish. Examples of equations include:

1 / 5. A fifth, same as 0.2.

3.1 + 2.5 * 3. As usual, multiplication takes place before addition.

(2 - 1) * (3 + 1). Brackets may be used to force operator
precedence the way you want.

2 ^ 7. 2 to the power of 7.

2 + 1 / tau. "tau" or "g" may be used to represent the golden ratio
(1.618034...). You may also use "pi" (3.14159...).

sqrt(2). Square root of 2. Other functions available include sin(),
cos() and tan(), which expect radians as input, or their equivalents expecting
degrees: sind(), cosd() and tand().

1 + 3r2. "r" may be used as shorthand for sqrt(). Brackets are not
required if taking the square root of a simple number. Also, multiplication is
presumed when "*" is left out, so "3r2" may be used instead of "3 * r2".

Most operations can be undone and redone, including changing to a new model,
changing the scale of the model, and changing face colors and images. Use
"Edit→Undo" (keyboard shortcut: Ctrl+Z) to undo, and
"Edit→Redo" (keyboard shortcut: Ctrl+R) to redo. Both
operations are also available on the main toolbar. If you
Right-click on either button, you'll see a menu of the last 20
operations that can be undone/redone. Select one to perform multiple
undos/redos in a single step.

See also "Edit→Undo Settings" for various options that control
the undo mechanism. You may limit the amount of memory used, and the number of
undo levels available.

Expand any view temporarily to become full-screen using F2 or
"View→Full Screen". The active view then takes over the screen and all
menus, toolbars and borders are hidden. It can be nice to view models this way
with no other distractions. To exit this mode, hit F2 again, or hit
Esc. Any operation that opens a dialog box will
also force an exit from full screen mode (e.g. opening a file with
Ctrl+O).

Geometry may be viewed in true stereoscopic 3D! To enable this, select from
the "Display→3D Stereo Display" submenu. Two slightly different
images will be rendered, one for each eye, giving the model a 3D appearance.
If you load a background image ("Image→Load Background Image"),
it will appear to be pushed back behind the model.

If you have red/blue or red/green glasses, you may use those. Even with
red/blue glasses, try the "Left Red, Right Cyan" option as it may work
better than the "Left Red, Right Blue" option. If you're after some
red/blue glasses, there are some available for under $3 including shipping
here.

The other option is to display the two images for your left and right eye
side-by-side. You have the option of putting them either way around. This
option has the advantage of maintaining full-color images and not requiring
glasses, but requires the user to be able to blend the two images themselves,
either by going slightly cross-eyed, or by allowing their eyes to drift apart
slightly. For the cross-eyed option, try holding a finger to the screen,
pointing up, with the tip between the two images, then pull the finger slowly
towards yourself. Keep your eyes focused on your finger but be aware of the
two images behind it. There will be a point where they overlap behind your
finger. Now try to keep those images fused and slowly take your finger away.

Personally I find the parallel-eyed version much easier to achieve and to
keep focused. For this, try to look past the screen into the distance and
focus far away. Then try to look down to the screen while keeping your eyes
focused in the distance. Again it is a matter of then trying to merge the two
images.

When changing from one polyhedron to another, the transition may be animated,
and there are plenty of ways to customise this. You can control what kind of
transitions is used in different situations, as well as adjusting parameters
governing how each type of transition behaves.

Enable Transitions: Use this to enable or disable transitions. You
may want to disable them in order to race through polyhedra more quickly.
However you might want to consider reducing the default transition time from
half a second to something even faster instead. More snappy, but still has a
nice feel to it.

Transition Options: Opens the Transitions dialog box.

The Transitions dialog box has a great many options for customising
transitions, but you needn't understand them all. If you just want to change
the type of transition, or the length of each transition, then make sure
"Occasion" is set to "Default", and change "Method" or "Duration" accordingly.
See below for a list of methods available.

The Transitions dialog box is modeless, meaning you can leave it open and
still use the rest of the program. This lets you see what effect changing
settings has without closing the window.

If you want to get into more detail, here's what the Transition dialog box
provides:

Setup: The complete set of transition options may be saved and
reloaded later. Choose an existing setup from the drop-down list, or enter the
name of a new setup. A number of predefined setups are built in. The
selection has no effect until you click one of the following buttons:

Occasion: Choose the occasion whose transitions you wish to edit.
This will populate the remaining settings to reflect the current settings for
that occasion. A line of text appears below here with a helpful note about the
selected occasion. Occasions to choose from are as follows:

Default: These settings are used for any occasion that doesn't
specify its own non-default settings.

Selected tour events: This lets you change the transitions for
one or more selected tour events.
See Tours below for information about tours.

Start-up: This transition only occurs when first starting the
program.

Load built-in model: Transition occurs when a
built-in model is loaded, e.g. when clicking the
right-pointing green arrow to move to the next model. If you hit the
left-pointing green arrow to move to the previous model, the inverse
transition is used. For example, if the transition normally moves the new
and old models to the left, then the inverse transition moves them to the
right. This gives a more intuitive feel as you move forward and backward
through the list.

Load from file: Transition occurs when loading a model from a
file. Inverse transition is used when moving to previous file in a folder.

Load from memory slot: Transition used when loading a model from
a memory slot.

Create new model: Transition used for various situations where a
new model is created, such as
creating a block, torus, 4D prism, duoprism, antiduoprism or 4D step prism.

Add base to dual/mirror/memory: Transition used when adding two
models together.

Convex hull/core: Transition used when creating a
convex hull.
The inverse transition is used when creating a
convex core.

<Multiple occasions...>: Allows you to set transition
options for multiple occasions at once. Opens a list of occasions where
you can tick any combination. Options that currently vary between
occasions will show <No change> until you set them
otherwise, and their drop-downs will show all the individual current values
for quick selection.

Method: Select the method to use for transitions. All settings from
here on apply to the current occasion only, as set above. Each transition
method has a few settings to tweak its behaviour. The methods are as follows:

No transition: Instant change.

Random: Select a transition type at random.

Sideways: The old model spins off one side while the new model
spins on from the other side. Settings are:

Angle: Angle of movement. Zero represents movement from
right to left, 90 is top to bottom, 180 right to left etc.

Fade-out spin: When non-zero, then model will spin faster
and faster as it moves off the screen, ending with the rate set here
just before it vanishes. The rate is given in full rotations per
second.

Orbit: The old and new models orbit each other as one grows and
the other shrinks. Has the following settings:

Tilt angle: Tilt angle of the orbit axis.

Number of orbits: Number of complete orbits around each
other.

Fade-out spin: Same as for the Sideways method above.

Shrink & Grow: One model shrinks in-place while the other
grows. Just one setting:

Fade-out spin: Same as for the Sideways method above.

Explode & Grow: Old model explodes while the new model grows
from inside. See below for settings.

Shrink & Implode: Old model shrinks while new model implodes
to enclose it. See below for settings.

Explode & Implode: Old model explodes while new model
implodes into place. Settings for this and the above two options are as
follows:

A tour is an animated polyhedral slideshow. For example, you may start
with a cube folding and unfolding, then switch to a dodecahedron morphing into
its dual and back, and so on. You may choose the transition to use between
each event, and may even transition between 3D and 2D views.
Tours may be played back or exported as a video.

Tip: You can stop a tour from playing back by hitting
Esc (or by hitting the play button again).

A tour consists of a list of events. Each event may have its own model,
view layout, and transition to the next event.
Tours can be saved and reloaded from .tour files.

The Tour menu provides various controls for creating and editing
your tour. The Tour toolbar provides another way to access all these same
controls. The menu contains the following:

Show/Hide Tour List: Show or hide the tour list, which lists all
events in the current tour. More on this below.

Open Tour: Browse to open a tour file.

Merge Tour: Find a tour file to merge into the current tour.
Events from the tour file are appended to the end of the current list.

Save Tour: Browse to save the current tour.

Play Tour: Play the tour. While playing a small play-symbol appears
next to the mouse pointer. Use this control again to stop playing the tour, or
hit Esc. The tour also stops playing if you click
on another item in the tour.

Add to Tour: Add the currently loaded model and view layout to the
tour list. It is added at the end, but you may then drag it in the list to any
position you want.

Replace Current Event: Replace the current event with the current
model. For example, you might select an event, make changes to how fast it's
spinning or folding, then save these changes back over the original event.

Remove from Tour: Remove the currently selected event or events from
the tour.

Previous Event: Step to the previous event in the tour.

Next Event: Step to the next event in the tour.

The tour list shows all the events in a list, one per row. You may drag
items within the list to rearrange them. You may click on an event to select
it, and use Ctrl+Left-click to select multiple events. The list has
columns, each showing different information about each event, including:

Name: Name of the tour event. After selecting, click in this field
again to edit the name.

Duration: Duration of event in seconds, not including transition
time between events. Click again once selected to edit.

Fade: Duration of transition between this event and the next. If it
shows <Default>, then the default transition duration is used.
Click again once selected to edit. Enter a time in seconds, or -1 to use the
default.

Fade Type: The transition method to be used between this event and
the next. Click again once selected to choose another method.

Inherit Layout: When "No", the view layout and rate of spinning,
folding, morphing etc are all loaded from this event. When "Yes", all this
information is inherited. Use "Yes" when you want to go from one event to
another but keep the same rate of rotation and same orientation etc.
Click to switch between "Yes" and "No".

You may Right-click on the tour list to open a menu with a few
relevant options:

Set Duration: Set the duration of the selected events.
This is more powerful than editing the duration in-place in the list, because
you may select multiple events first.

Transition Options: Open the Transition dialog box (see
above) and set the occasion to "Selected tour
events", allowing you to edit all the transition options for one or more tour
events.

Inherit Layout: Switch the inherit-layout flag on or off for all
selected events.

Export Video: Open the
Exporting Image/Video dialog box, and automatically
select the option to export the current tour as a video.

Overall display of vertices and edges may be enabled or disabled. They may
be shown either as points and lines, or spheres and cylinders respectively.
All settings are remembered after you exit the program for next time.

Show Vertices(keyboard shortcut: V).
Enable this option to display a small point or sphere at each vertex of the
model.
Note: only true vertices are shown, not virtual ones where faces
intersect.

Show Edges(keyboard shortcut: E).
Enable this option to display a line or cylinder along each edge of the
model.
Note: only true edges are shown, not virtual ones where faces
intersect.

Show Stellation Vertices(keyboard shortcut: Shift+V).
Enable this option to display a small point or sphere at each vertex of the
stellated model. The default stellation for a model is the model itself,
except that it only has the externally accessible parts as faces, so for
the base model this lets you see all the virtual vertices too.
(See Stellated Polyhedra).

Show Stellation Edges(keyboard shortcut: Shift+E).
Enable this option to display a line or cylinder along each edge of the
stellated model. The default stellation for a model is the model itself,
except that it only has the externally accessible parts as faces, so for
the base model this lets you see all the virtual edges too. It is
also useful for seeing a wire-frame of some other stellated model, with the
solid core inside. (See Stellated Polyhedra).

Size for Drawing Vertices. Allows you to adjust the size of the
points displayed at each vertex.Tip: Right-click on the Show Vertices toolbar button to
open a small menu where you can quickly choose this option.

Line Width for Edges. Allows you to adjust the width of lines
displayed along each edge.Tip: Right-click on the Show Edges toolbar button to
open a small menu where you can quickly choose this option.

Use Spheres and Cylinders. Select this option to display
spheres and cylinders rather than points and lines at vertices and
edges respectively.

Sphere and Cylinder Options. Opens
a window allowing control of various options for displaying spheres and
cylinders. The window is modeless, so you can still use the rest of
the program while this window is open, and immediately see the effects of
your changes. You may specify the material used (solid color, same as
points/lines, gold, silver, copper, wood or stone). You may also set the
radius of the spheres and cylinders. The cylinder radius is set as a
fraction of the sphere radius (e.g. 0.5 for half the radius). The sphere
radius is generally also set as a fraction of another value. That value
may be chosen from the following choices:

Inverse square root of number of edges. This may sound like
a somewhat arbitrary choice, but I found it through much
experimentation and it works very well for a wide range of models.
Generally, the more edges there are, the smaller you want the spheres,
which is what this achieves. For this reason it is the default
setting.

Edge length of model. You may want the cylinder radius to
be proportional to the edge length.

Radius of model. Here the sphere radius is proportional to
the physical size of the model, but not the complexity.

Specify actual measurement. Rather than a fraction of some
other measurement, this allows you to explicitly enter the sphere
radius.

Tip: Right-click on the Use Spheres and Cylinders toolbar
button to quickly open this options window.

The Display menu includes a group of items for hiding or showing
individual cells, faces, edges or vertices. Most of these require you to
select a cell, face, edge or vertex first (see
Selecting Faces, Edges and Vertices).

Tip: Vertices and their dual faces are hidden in sync. So when you hide
an individual vertex, it will hide the matching dual face too.

When a cell, vertex or edge is selected, the text in the menu items
below will have "Face" replaced by
"Cell", "Vertex" or "Edge" as appropriate.
The relevant items from the Display menu are as follows.

Show/Hide Faces of Selected Color(keyboard shortcut: H).
The selected cell/face/edge/vertex is hidden along with all others
of the same color. If any were already hidden, then all are unhidden instead.

Show/Hide Faces of Selected Type(keyboard shortcut: Shift+H).
The selected cell/face/edge/vertex is hidden along with all others
of the same type. If any were already hidden, then all are unhidden instead.

Show/Hide Selected Face(keyboard shortcut: Ctrl+H).
The selected cell/face/edge/vertex is hidden. If it was already
hidden, then it is unhidden instead.

Show/Hide All Faces(keyboard shortcut: Ctrl+Shift+H).
All cell/faces/edges/vertices are hidden. If any were already
hidden, then all are unhidden instead.

Toggle Shown/Hidden Faces(keyboard shortcut: Ctrl+Shift+T).
All hidden cell/faces/edges/vertices become unhidden, and all those
unhidden become hidden. This is useful when you want only a few faces
displayed: start by hiding the faces of interest, then toggle so that all other
faces are hidden instead. Similarly, unhiding a specific face can be tricky
since you can't select it with the mouse. Start by toggling all hidden faces,
then select the face of interest and hide it, then toggle again.

Show/Hide Faces in Selected Plane. Hide all faces in the same plane
as the selected face. If they are all hidden already, then they are all
unhidden. This option applies to faces only.

Show/Hide One Part of Compound. Hide one whole part of a compound
(the one with the selected cell/face). If already hidden, it is
unhidden instead.

Show/Hide Faces Around Vertex(keyboard shortcut: Ctrl+Shift+V).
All faces around a specific vertex are hidden. If any were already hidden,
then they are all unhidden instead. A window appears allowing you to enter the
index of the vertex to use. If you select a vertex first with the mouse, then
the index will default to that vertex, so you can just click OK or hit
Enter.

Hide Edges & Vertices Between Hidden Faces. This item is a
setting which may be enabled or disabled, rather than a one-off action. When
enabled, any edges or vertices surrounded only by hidden faces will also be
hidden, regardless of whether they have otherwise been flagged as hidden.

Note, the hidden attribute of a cell/face can also be copied
quickly between cell/faces using
Color/Hide Faces Mode.

Models may be colored in various ways. This section describes the items on
the Color menu. See also
Color/Hide Faces Mode for rapidly spreading colors
across selected faces.

Rainbow Color Mode(keyboard shortcut: Shift+R).
This turns Rainbow mode on and off. Rainbow mode overrides all
other color settings. It draws models in white only, but uses three
lights in the colors red, green and blue. This makes the faces change to
all sorts of colors as a model is rotated, and makes it easy to
distinguish between faces. It is similar to the coloring style used by
the program Mathematica.

Basic Color Scheme. This submenu lets you choose the basic
color scheme that will be used to color the faces. There are three
groups of options here:

Exactly one option at a time may be selected from the first group.
With the exception of "Auto Color", they are listed roughly
from the least to the most colorful.

Auto Color. This is the default setting.
Firstly, if the model is a compound, then each component is given
its own color. Otherwise, the polyhedron is colored according to
face type. If there's only one face type, a different color is
used for the front and back of faces, to make it more colorful. If
there are greater than 5 face types, chiral pairs are colored the
same way to reduce the number of colors and highlight reflective
symmetry.

Use a Single Color. A single color is used for all
faces of the model.

Color Faces by Number of Sides. All triangles will use
one color, all quadrilaterals another color, all pentagons another
color and so on. For example, there are two different types
of triangle in the snub cube, but they are both given the same
color using this method.

Color by Face Type. Each type of face is given its own
color. Faces are of the same type if they fit into the
model the same way. The snub cube, for example, has two different
types of triangle faces. One has another triangle across each
edge, and one has a square across one of its edges. So these would
be colored differently using this method. The next group of
options below can be used to tweak this behaviour.

Color Along Cross-Section Direction. Faces within each
type are colored differently depending on how far along the
cross-section direction they occur (this is an axis orthogonal to
the cross-section plane). The main purpose of this is to make
cross-sections more colorful, while still retaining their own color
symmetry. See Cross-Sections.

Color per Face (Unless Parallel). Every face is given
its own color, unless they are parallel. All parallel faces are
given the same color with this method (even if they are not the
same shape).

The second group of options may be enabled independently. They are
all available when using "Color by Face Type" from the first
group of options above. Otherwise some options will not be available,
as appropriate.

Same Color for Chiral Face Pairs. Faces that are
mirror images of each other will be colored the same.

Same Color for Coplanar Faces. Coplanar faces will
always be colored the same.

Same Color for Coplanar Chiral Pairs. Faces that are
coplanar and mirror images of each other will be colored the same.

Same Color for Front and Back of Faces. The front and
back of each face will be colored the same.

Special Color Arrangements. This submenu lets you choose a
predefined color arrangement, for faces with certain symmetry. You may
choose one arrangement for each kind of symmetry, or select it again to
disable it. Any settings here will override the basic color scheme set
above (for the relevant faces only). The first dodecahedral and first two
icosahedral arrangements are as described in Magnus Wenninger's
Polyhedron Models.

Dodecahedral Arrangement 1 (4 colors).

Dodecahedral Arrangement 2 (6 colors).

Icosahedral Arrangement 1 (5 colors).

Icosahedral Arrangement 2 (5 colors).

Icosahedral Arrangement 3 (10 colors).

Rhombic Triacontahedral Arr 1 (5 colors).

Rhombic Triacontahedral Arr 2 (15 colors).

Cubic Arrangement (3 colors).

Octahedral Arrangement (4 colors).

Rhombic Dodecahedral Arr 1 (3 colors).

Rhombic Dodecahedral Arr 2 (4 colors).

Rhombic Dodecahedral Arr 3 (6 colors).

For example, to color the truncated dodecahedron, and avoid any faces of
the same color sharing an edge, you might set the basic color scheme to
"Auto Color" and enable the first dodecahedral arrangement. Since
we haven't chosen a special icosahedral arrangement, the icosahedral
triangles would all be the same color, which is fine because they don't
share edges with each other. The dodecahedral arrangement however, will
use four different colors for the decagrams, ensuring that decagrams of the
same color will never share an edge.

Replace Face Color Throughout...(keyboard shortcut: C).
Change the color of all faces with the same color as the selected face, in
both the base model and the dual. Backs of faces which are the same color
are also changed. The user is prompted for a new color.
In 4D, applies to cells rather than faces.
If an edge or vertex has been selected, then the operation applies to edges
or vertices instead. Note: individual coloring of vertices is only
possible when "Color→Color Vertices Same as Dual Face" is
enabled.

Set Face-Type Color...(keyboard shortcut: Shift+C).
Change the color (front and back) of all faces of the same type as the
selected face.
In 4D, applies to cells rather than faces.
If an edge or vertex has been selected, then the operation applies to edges
or vertices instead. Note: individual coloring of vertices is only
possible when "Color→Color Vertices Same as Dual Face" is
enabled.

Set Single Face Color...(keyboard shortcut: Ctrl+C).
Change the color of the selected face only. If the back of the face is the
same color as the front, then it is also changed.
In 4D, applies to cells rather than faces.
If an edge or vertex has been selected, then the operation applies to that
edge or vertex instead. Note: individual coloring of vertices is only
possible when "Color→Color Vertices Same as Dual Face" is
enabled.

Set all Face Colors...(keyboard shortcut: Ctrl+Shift+C).
Change the color of all the faces, front and back.
In 4D, applies to cells rather than faces.
If an edge or vertex has been selected, then the operation applies to edges
or vertices instead. Note: individual coloring of vertices is only
possible when "Color→Color Vertices Same as Dual Face" is
enabled.

Replace Back-Face Color Throughout...(keyboard shortcut:
B). Change the color of all backs of faces with the same color as the
back of the selected face.

Set Back-Face-Type Color...(keyboard shortcut: Shift+B).
Change the color of the backs of all faces of the same type as the selected
face.

Set Single Back-Face Color...(keyboard shortcut:
Ctrl+B). Change the color of the back of the selected face only.

Set all Back-Face Colors...(keyboard shortcut:
Ctrl+Shift+B). Change the color of the backs of all the faces.

Take Color from Neighbouring Faces. All faces of the selected
face color will receive a new color from one of their neighbouring faces.
Neighbouring faces with the original color are ignored, but otherwise some
options are presented if two neighbours have different colors. If any
faces are completely surrounded by faces of the original color, the process
is repeated until no further changes occur. This is quite an obscure
operation, see my tutorial on
hollow spherical
models for an example of its use.

Set Color of One Part in Compound. Set the color of one whole
component within a compound. Select a face first to indicate which
component is intended.

Color as a Compound(keyboard shortcut: Shift+A). Color
each component of a compound differently. If a stellation view is
selected (see Stellated Polyhedra),
then try to recognise the current stellation as a compound of
simpler convex polyhedra, and use a different color for each part. This
works well except for the case where faces from different parts are
coplanar. For example, five octahedra and ten tetrahedra won't work, but
two tetrahedra, five tetrahedra, or five cubes work well. This is only an
issue when acting on a stellation view though. If the base model itself is
a compound, then there is no ambiguity and it will be colored correctly.

Color Along 4D Projection Axis. Cells within each type are
colored differently depending on how far along the 4D projection axis they
occur. The main purpose of this is to make 4D projections more colorful,
while still retaining their own 3D color symmetry.
See 4D Projection. (Only available for 4D models).

Stellation Diagram Color Scheme. This submenu lets you choose
a scheme for coloring the stellation diagram, which appears in either the
stellation diagram view, or attached to the selected face when
"Selection→Selected Face Display→Show Stellation Diagram"
is selected (see Stellated Polyhedra).

Color Per Front/Back/Internal. Use red for front-facing
regions, yellow for back-facing regions, green for internal regions,
and blue for external regions.

Color Per Front/Back. Use red for front-facing
regions, yellow for back-facing regions, and do not fill other regions.

Color Per Facet. Regions belonging to the same
facet are shown in the same color. That is, regions that
are equivalent under symmetry transformations.

Color Per Chiral Pair. As above, but regions that are
mirror images of each other are also shown in the same color.

Color Per 2D Layer. Each layer of regions within the
stellation diagram is colored differently.

Color Per 3D Layer. Regions are colored according to which
3D layer the cell below belongs to.

Vertex Colors. Submenu for control over vertex colors.
Tip: Right-click on the Show Vertices toolbar button to open a
small menu where you can quickly choose these options.

Default Vertex Color.... Change the default color used to
display vertices when shown as points, or when displayed as spheres if
sphere color has been set to match point color. To control the color used
when shown as spheres, use "Display→Vertex
& Edge Options→Sphere and Cylinder Options".

Color Vertices Same as Dual Face. When ticked, vertices may be
colored individually, their colors being kept in sync with their dual
faces. Thus changing a vertex color will also change the matching dual
face color. In this case, the default vertex color is ignored.

Edge Colors. Submenu for control over edge colors.
Tip: Right-click on the Show Edges toolbar button to open a
small menu where you can quickly choose these options.

Default Edge Color.... Change the default color used to display
edges when shown as lines, or when displayed as cylinders if cylinder color
has been set to match line color. To change the color used when shown as
cylinders, use "Display→Vertex & Edge
Options→Sphere and Cylinder Options".

Color All Edges the Same. The remaining options on this submenu
represent different coloring schemes for edges. Exactly one will be active
at any given time. This first option simply colors all edges the same,
using the default edge color.

Color Edges by Type (including reflections). Each edge type is
given its own color. Edge types which are mirror reflections of each other
are also colored the same.

Color Edges by Type (excluding reflections). Each edge type is
given its own color, but edge types which are mirror reflections of each
other are colored differently.

Color Edges Same if Parallel. Edges are colored the same only if
they are parallel.

Color Edges by Great Circles. Edges are colored the same only if
the lie in the same planes through the centre of the model.

Color Edges from Neighbouring Faces. Edge colors are set by
averaging the colors of connected faces.

Color Edges from Neighbouring Faces (complement). Edge colors
are set by averaging the colors of connected faces, and taking its
complement, making them stand out better against those faces.

Set Midsphere Color.... Change the color used to display the
wire-frame sphere when "Display→Show Midsphere/Reciprocation
Sphere" is enabled.

Set Cell Line Color.... Change the color of cell lines as
displayed using "Stellation→Show Cell Lines".

Set Background Color...(keyboard shortcut: G). Change
the background color. The user is prompted for a new color.

Use "Display→Polygon Filling Options..." to open a dialog box with
various options for polygon filling. It is a modeless dialog box,
meaning that you can leave it open while you use other features of
Stella4D.

An image is displayed which shows a preview of the current settings. See
how it changes as you select different options. The following options are
available.

How to color areas where coplanar faces overlap. The options in
this section decide what color to use in areas where coplanar faces of
different colors overlap (such as the compound of 10 tetrahedra).

Blend colors. The colors from all overlapping faces are averaged
to obtain a new color.

Face with smallest area on top. Use the color of whichever face
has the smallest area, or pick one at random if they have the same area.
This ensures that if one face lies entirely within another, it will still
get its own color.

Single new color. When two or more faces of different colors
overlap, use a single new color. Click the "Color" button to select that
color.

Filling method. These options decide which areas within a face
should be filled and which should be left empty (consider for example the
octagram faces of the great rhombihexahedron).

Auto (depends on orientability). Use the density method for
orientable polyhedra, or the modulo-2 method otherwise.

Density/winding method. Only areas wound by a face's edges a
total of zero times will be left unfilled. Areas wound more than once will
be given a higher density (for example, the centre of a pentagram will have
density 2).

Modulo-2 method. Any area wound an even number of times will be
left unfilled (the centre of a pentagram will be left empty by this
method).

Allow coplanar faces to cancel each other out. If two faces overlap,
but are wound in opposite directions, they may cancel each other out leaving an
unfilled area.

Color dense areas within a face more densely. This alone may not
make a difference, for example the centre of a pentagram will usually be
colored the same as the lower-density arms. But it can make a difference if
the pentagram overlaps another face, or if either of the following two options
is set.

Lighten dense areas. The more dense an area becomes, the more it is
lightened, according to the percentage set.

Darken dense areas. The more dense an area becomes, the more it is
darkened, according to the percentage set.

Note: cross-sections will always use the "Auto" filling
method, and always allow coplanar faces to cancel each other out, regardless of
the above settings. Otherwise some areas may be filled incorrectly.

Images may be put on faces. You may want photos of your pets on the faces
of a dodecahedron, or you may want a model to look like it's made of wood.
This section describes the items on the Image menu (see also
Image mode below).

Load Image...(keyboard shortcut: Ctrl+I).
Load an image from a file to display on the selected face. A file browser
appears allowing you to find the image you want. Select an image in the
browser to see a preview of the image along with its dimensions. Once an
image is opened, a window with various image options. This window is
similar to the one available with Image Options below.

Load Image List.... Load a text file which contains a list of
image files to load. Select a text file in the browser to see a preview of
the first couple of lines. The file contains one file name per line. File
names are relative to the text file, or full paths can be given. Blank
lines are ignored, and comments may be included in the file after "#".
Each image is applied to a single face. Hidden faces are skipped. A
window with image options appears, allowing the user to set options which
will apply to all images loaded. It also gives a choice about which faces
the images should be applied to.

Save Image List.... Save a text file containing the list of
images on faces of the current model.

Persistent Image List. When enabled, a loaded list of images
will be remembered and re-loaded onto any new model that's loaded.

Apply Image to Other Faces.... Apply the image on the selected
face to other additional faces. Gives the user the following options:

All faces of same type as selected face

All faces with same color as selected face

All faces with no image

All faces

Cancel

Another way to apply an existing image to new faces is to use Color Mode (see below).

Remove Images.... Remove images from some or all faces.
A list of options is presented to the user, similar to above.

Image Options.... Opens a window with various image options.
The options are also available directly elsewhere on this menu, so see
above and below for details of what the options do. This window provides a
convenient way to set multiple options at one time, and there is a preview
in the bottom right corner which shows you the effect of an option as you
move the mouse over it.Tip: Right-click on the Image mode toolbar button to open a
small menu where you can quickly choose this option.

Options for All Images.... Opens a window similar to above, but
options set here will be applied to all images on the current model. Only
set the options you wish to change.Tip: Right-click on the Image mode toolbar button to open a
small menu where you can quickly choose this option.

Image Projection. Submenu with various projection types for
mapping images onto a model.

Project onto each Face Separately. Image is projected onto
each face individually, as if a separate copy is lying flat on each
face.

Planar. Single image is projected through the whole model.
Works well for a wood-grain or marble image. It will appear stretched
on faces that are at a sharp angle to the image.

Cylindrical. Image is wrapped around model in a cylindrical
fashion. Less stretching at the sides, but there still may be
stretching for faces at the top and bottom.

Spherical. Image is wrapped onto a sphere and projected
inwards through the model. This has the least stretching, although the
wrapping onto a sphere itself may stretch the image near the top and
bottom if the image was not designed for spherical mapping.

Image Orientation. Submenu specifying what orientation to map
the image.

Align Height of Image to Selected Item. Use this one to
align the axis for cylindrical or spherical mapping to a selected face,
edge, vertex or rotational symmetry axis.

Fit Image Within Face. Resize and reposition the image so that
it fits entirely within the selected face.

Fit Image Around.... A submenu which lets you resize and
reposition the image so that it fits tightly around the selected face, over
the entire model, or over all those faces with the image applied.

Unstretch Image. An image can be stretched in Image
mode (see below). Use this option to
unstretch it again.

Flip Image Horizontally. Use this if your image is
back-to-front.

Keep Image Upright. Keeps images upright on faces regardless of
the orientation of the model. Only works when "Project onto each Face
Separately" is chosen from the "Image Mapping" submenu (see
above).

When Rotating, Snap Image to Edges. When rotating an image in
Image mode (see below), snap to
orientations where an edge of the image is parallel to an edge of the
selected face. If "Keep Image Upright" is enabled (see above),
then rotation snaps when image is close to landscape or portrait instead.
When this option is disabled, you may freely rotate to any angle.

Show Whole Image in Image Mode. When in Image mode
(see below), show a transparent version of the
entire image, even though it may extend off the edge of the model. This
makes it easier to see what part of the image is on a face, and how close
to the edges you are etc. Disable if this is distracting. When a face is
selected, feint lines are also drawn on the image to indicate the face
mapping.

Blocky Pixels when Close-Up. When ticked, you may zoom in to
see an image's sharp-edged pixels. Otherwise, the pixels will be smoothed
(the default).

Image Boundaries. A submenu which lets you choose what happens
outside the edges of the image. An endless border of any color may be used
around the image, or the image may be repeated endlessly, or values can
be clamped to the edges of the image.

Use Face Color in Addition to Image. Normally when you put an
image on a face, you don't also want the original color of the face to show
through, but enable this option if you do want it to show through. You may
create some interesting effects by mixing a color with your image.

Load Background Image. Opens a file browser, allowing the user
to find an image which should be displayed in the background. Select an
image in the browser to see a preview of the image in the browser.

Remove Background Image. Stop showing an image in the
background.

Background Image Fit. Submenu for choosing how background image
should fit within each view.

Fit Within. Background image is scaled to just fit within
each view. Space left at the top or sides is filled with the usual
background color.

Fit Around. Background image is scaled to just fit around
each view, leaving no space uncovered. This may crop the image at the
sides or top.

Fit Width. Background image is scaled so that its width
matches the width of each view. The top and bottom may be cropped,
or space may be left which is filled with the usual background color.

Fit Height. Background image is scaled so that its height
matches the height of each view. The sides may be cropped, or space
may be left which is filled with the usual background color.

Stretch. Background image is stretched to fit the width and
height in each view. This is the only option where the image's aspect
ratio is not maintained.

Actual Image Size. Background image is shown at actual size
(pixels of the image mapping to pixels on the screen). The image may
be cropped, or space may be left around it, filled with the usual
background color.

Show BG Image Through Model. Allows the background image to
show partially through the model, which may be handy if you're aligning
something. The value must be between 0.0, where the image won't show
through the model at all, and 0.95, where the image almost completely hides
the model. A value of 1.0 would hide the model completely which could lead
to confusion, so the 0.95 limit has been set.

Maximum Image Size. Images can take up a lot of memory,
generally a lot more than the original size of the image file on your hard
drive (because it is no longer compressed). This option lets you limit the
size used internally for images. Digital cameras often create very large
images, but you won't see any difference on screen if smaller images are
used. You only need to tweak this value in two cases:

Make it smaller if you are running out of graphics memory due to
lots of large images.

Make it bigger if you don't think the resolution is high enough on
printed nets and pictures (printers are much higher resolution than
screens).

Text may be displayed on any face, edge or vertex. The text may include some
limited HTML markup, and may sit in various ways. Text on faces will also
appear in nets. Make your own polyhedral dice with numbered sides!

Options for dealing with text may be found under the "Edit" menu as
follows:

Text for Face/Edge/Vertex(keyboard shortcut: Ctrl+T).
Allows you to enter the text you want to appear on the selected face, edge or
vertex (select an item first). Details on format below.

Load Text List. Submenu allowing you to browse to a text file.
Each line of the text file will be put on a different face, edge or vertex.
Hidden faces will be skipped.

Persistent. Tick this option if you want the text list to be
loaded again onto each new model you load.

Save Text List. Submenu allowing you to save the current collection
of text on faces, edges or vertices to a text file.

Remove Text. Allows you to remove text from the model in a bunch of
different ways.

Text Options. Opens a dialog box with various options for how text
should appear on the model. Text may be aligned to the screen, or remain flat
on a face, or extend out from a vertex, or stand along an edge, and so on.

Text may span multiple lines, and may include some limited HTML. The following
HTML is supported:

Links: <a href="link">link text</a>.
Follow the link by double-clicking on it or using Shift+Left-click.
The link may take the following forms:

Link to a website:
<a href="http://www.software3d.com/Stella.php">Stella</a>

Send an email:
<a href="mailto://someone@domain.com">Email</a>

Open some random file:
<a href="file://Blah.txt">Open Blah.txt</a>

Open a built-in model. Uses custom identifier
"shape:". Note: you can use "shape://", but the "//" is optional.
Name of shape can be anything you would type into the
search box, and will open the first shape that would
be found that way. E.g.<a href="shape:d">Dodecahedron</a><a href="shape:cubes 5">Compound of 5 Cubes</a>

If file name after "file://" is recognised as a Stella4D file,
that file will be opened within Stella (it won't open a second instance of
Stella). You can use this with .stel files and
.tour files. Paths may be absolute, or relative to
the currently open file:<a href="file://Shape.stel">Shape</a><a href="file://Demo.tour">Take the demo tour!</a>

This means you could pretty much put together a polyhedral
website or file browser now. However this would only be usable inside
Stella, as there's currently no way to export it as an interactive web page.

Use the custom "noColor" to stop links from being colored
blue: <a noColor href="shape:cube">Cube</a>

The <else> part is optional. You may put other HTML tags inside too.
You can even split bookend tags between separate <sel> tags. For
example, to make text bold when a face is selected:<sel><b></sel>Blah!<sel></b></sel>

You can use this to hide links till you select a face. E.g.,
just make a big "+" appear when selected:<sel><font size=+4><a href="http://www.google.com">+</a></font></sel>

Rendering in Geomag style makes polyhedra appear as if they were built using
the Geomag magnetic construction
kit. Generally Geomag-style rendering will be permitted unless a model
could not be physically built using Geomag, in which case a reason will be
shown on the screen. Some common polygons not available from Geomag are
subdivided using ones that are.

Subdivide Inwards. Geomag do not make octagons and decagons, so
cupolae are used in their place, constructed from squares and triangles.
However these are not flat, so this option decides whether such cupolae should
poke out or poke in.

Geomag Edge Style. Submenu allowing you to select an edge style from
those available in the Geomag construction kit.

When selecting face colors, you will be limited to those available for Geomag
panels.

If images are applied to the faces, these will appear
as if using Geomag's Deko panels. When printing the unfolded net view, only
the parts you require for use with the Deko panels will be printed, ready to
cut out and insert into the Deko panels.

A collection of models suitable for construction with Geomag can be found in
the Geomag Library folder within the Stella
library.

Select a view and use "View→View Diagrams→Cross-Section" or
the matching toolbar button to switch to the Cross-Section view. This shows
a 2D cross-section (or slice) through the current polyhedron (or 3D cross-section if the current model is a 4D polytope).
The cross-section is made with a slicing plane, which can be controlled
in various ways. Any edge of the original model passing through the plane is
sliced to become a vertex. Any face crossing the plane is sliced to become one
or more edges (nonconvex polygons can lead to more than one edge). And a whole
polyhedron is sliced to create one or more polygons. So each entity loses one
dimension as a result of the slicing process.
When slicing a 4D polytope, a hyperplane is used, each cell becomes one or more
faces, and the whole polytope becomes one or more polyhedra.
Note: many edges and faces may lie entirely on one side of the slicing plane,
and so do not contribute at all to the cross-section.

Cross-sections have a beauty all of their own, especially when animated by
altering the slicing depth (keeping the slicing plane parallel, but
moving it through the model). The slicing depth is a value between 0.0 and
1.0, each representing a plane at opposite ends of the model.

Each polygon of the 2D cross-section is surrounded by edges formed by
slicing faces. The edges are shown in the color of their corresponding face,
and the polygon itself is filled with a color obtained by averaging the colors
of its surrounding edges, weighted by their edge-lengths. This can produce
some pleasing results, with the polygon colors changing smoothly into other
colors as their edges get longer or shorter. When multiple polygons overlap,
the overlapping colors are also blended, making even more interesting images.

More colorful cross-sections can be obtained by using "Color→Basic
Color Scheme→Color Along Cross-Section Direction", especially for
regular polyhedra, which would normally be shown all in one color. When all
faces have the same color, the detail of the cross-section can be lost since
all edges and filled regions become the same color. With this option,
cross-sections become much more colorful, but still retain full color symmetry.

In 4D the case is a little different. Since cells are colored rather
than faces, the cell colors are used directly as face colors in the
cross-section. Overlapping faces can still have their colors blended however
(try the polytope known as "idfix" for example. That is, hit Ctrl+N, type
"idfix" and hit Enter).

The mouse may be used in the following ways.

Shift+Left-click: click on an edge to select the face it
represents.

Shift+Right-click: same as Shift+Left-click.

Ctrl+Left-drag: change the slicing depth. Mouse inertia
is supported as usual, so let go of the Left button while still dragging the
mouse and the animation should continue on its own at the current speed.

Ctrl+Right-click: set the slicing depth to exactly
half-way. This is often a point of particular interest.

Note: Shift+Left-click and Shift+Right-click act differently
for a 4D model. See 4D Cross-Sections for details.

The Section menu offers further cross-sectioning options:

Cross-Section Direction submenu:Tip: Right-click on the Cross-Section Tumble Mode or the
View Cross-Section toolbar buttons to open a menu with these same
choices.

Selected Face/Edge/Vertex/Symmetry First. Use the selected
face/edge/vertex/symmetry axis to decide the slicing direction. For a
face, the slicing plane will be parallel to the face. For a vertex, the
slicing plane will be orthogonal to a line through the vertex and the
centre of the polyhedron.

Along N-fold Axis. Slice perpendicular to the
N-fold symmetry axis. N changes depending on the symmetry
group of the model. Three such menu items appear, for the (up to) three
different types of rotational symmetry present.
(3D models only).

Cell First. Use a slicing hyperplane parallel to the current
cell. (4D models only).

Same as Projection. Lock the slicing direction to the 4D
projection direction. (4D models only).

Measured Item First. Slicing plane is defined by current item
defined in Measurement Mode (a point, line, or face). The slicing depth is
also set to align with the defined item (see
Measurement Mode).

Automatic. When ticked, choose whatever seems like the nicest
cross-section direction whenever a new model is loaded. This is generally
whichever direction will lead to a cross-section with the most symmetry.

Set Cross-Section Depth... Explicitly set the slicing depth to a
value between 0.0 and 1.0. As with all real number fields, you may also enter
an equation here.

Snap Sectioning Depth to Vertex. Change the slicing depth (up or
down) so that the slicing plane passes through the nearest vertex from the
original model.

Show Cross-Section in Model Views submenu:

Always. Always show the cross-section embedded in the model
views (i.e. Base, Dual, and Base + Dual views). It can be helpful to see
how the cross-section fits into the original model. When embedded in this
way, using Ctrl+Left-drag and Ctrl+Right-click in the
model views behaves as it would in the Cross-Section view instead, that is,
to change the slicing depth.

When Cross-Section View is Open. Only show the cross-section
embedded in the model views when there is a Cross-Section view open
elsewhere.

Never. This is the default. Don't ever show the cross-section
in the model views.

Show Vertices Near Slice Plane. When ticked, vertices of the
original model are displayed in the Cross-Section view when they get close to
the slicing plane. They are represented as white circles
(or spheres for 4D models)
which grow in size as the slicing plane approaches, then shrink again as it
passes by. They become green if they lie precisely in the slicing plane. It
is as if spheres were placed at each original vertex, and they too were being
cross-sectioned to create circles.

Fill 2D Cross-Section. Untick this option to show cross-sections
only as an unfilled outline. For extremely complex models, the filling can be
very slow, so untick this option if you find such a model. Outline-only
cross-sections can be displayed almost instantly for any model.

Show 2D Cross-Section Edges. Untick this option to show
cross-sections as filled areas only with no outline. Sometimes this is
aesthetically more pleasing, giving a "smoother" appearance.

The yellow left/right buttons in the title bar of the Cross-Section view
also serve a purpose. They skip the slicing depth forward or backward to all
the values where the slicing plane passes through a vertex of the original
model. These are often quite interesting points in the transition.

Use "Poly→Create Convex Hull" to create the convex hull of the
model in the current view. This is the smallest polyhedron which encloses all
vertices of the original model. You may even take the convex hull of a
partially folded net.

Use "Poly→Create Convex Core" to create the convex core of the
model in the current view. This is the largest polyhedron surrounding the
centre of the model which does not intersect any facial planes of the original
model (any planes through the centre of the model are ignored). The convex
core is the dual of the convex hull of the dual of the original model!

Use "Poly→Zonohedrify" (keyboard shortcut: Z) to create a
zonohedron based on the current polyhedron. A dialog box appears with various
options. A zonohedron is created from a seed star, that is, a set of
vectors which represent the directions of edges in the resulting zonohedron.
You may populate your star using combinations of vertices, edges, faces;
choosing for each whether you want all, all of the same type, or just a single
item. You may also include the current symmetry axes or world X/Y/Z axes in
the star. Note: all edges in the resulting polyhedron will have the same
length, even if vertices etc. were at different radii.

You may choose to either create a zonohedron purely from the chosen inputs,
or to add these zones to the existing polyhedron (to create a zonish
polyhedron). In the latter case you may also enter a scale for the new zone
edges. At 1.0 they will be the same length as the existing edges.

You may also specify the maximum number of zones to add. This may be handy
if you want to see the result of adding one zone at a time. Recreate the
zonohedron from the same initial model but increasing the maximum one at a time
to create successive steps in the zonohedrification.

Use "Poly→Subdivide Faces" to subdivide a model's faces into
smaller triangles. You will be prompted to enter the number of subdivisions
along each edge. Faces having more than three sides are first subdivided into
triangles, each meeting at a new vertex at the face's centre. You may choose
"1" as the number of subdivisions to just do this first step.

Tip: Be sure to display edges, or it may be hard
to tell that a face has been subdivided at all.

Subdividing faces can be useful in combination with
Faceting Mode. With more vertices spread across each
original face, you have more freedom to create interesting facets from them,
possibly carving interesting shapes out of the original faces. For some
examples of what this can achieve, see the following models:

Use "Poly→Create Geodesic Sphere" (keyboard shortcut:
Ctrl+G) to create a geodesic sphere based on the current model.
Traditionally geodesic spheres are based on an icosahedron, octahedron, or
occasionally a tetrahedron, but here any model may be used as a starting point.
Try starting with one of the Stewart Toroids for interesting results!

You will be prompted for the frequency of the geodesic sphere, which
is the number of subdivisions along each edge of the original polyhedron.
As with subdividing faces above, faces having more
than three sides are subdivided into triangles first. Once the faces are
subdivided, all vertices are projected onto a sphere.

Use "Poly→Create Waterman Polyhedron" (keyboard shortcut:
W) to create Waterman polyhedra. A dialog box appears with options for
different types of Waterman polyhedra. Once created, use the Left and
Right arrow keys to step through the sequence of Waterman polyhedra.
There is also an option to create 4D Waterman polyhedra. One interesting thing
about these is that their 3D cells are often Waterman polyhedra themselves, but
not always!

Use "Poly→Create X x Y x Z Block" to create a block where each
side consists of a grid of square faces. This may be a useful starting point
from which to augment or excavate cubes if creating
a large structure built from cubes.

A stellation of a polyhedron is a new polyhedron which has faces that lie in
the same planes as the faces of the original model. Typically you start with a
convex polyhedron and build stellations out from it, but the seed polyhedron
does not have to be convex. Stella4D is able to find all the
possible stellations of a given polyhedron.

For models which may be stellated (a small set in the demo, but not restricted
in the full version of Stella4D), use the Up and
Down arrow keys to visit each valid stellation (same as
"Stellation→Next Stellation" and
"Stellation→Previous Stellation", or the up and down green arrow
buttons on the options toolbar or at the top of a stellation view).
You can hold the keys down to see all the different stellations racing past.
Note however that you won't see anything happen in the base polyhedron view
though. The default views show the base polyhedron and its net. Instead you
need to change one of them to a view of a stellation, by selecting either the
View Stellation or View Dual Stellation buttons from the
view toolbar.

What counts as a valid stellation depends on your settings. The default
is that only fully supported stellations are counted as valid. Choose a
different criterion by selecting a different item from the top section of the
"Stellation→Stellation Criteria" submenu. This affects what
stellations you will encounter when hitting the Up and
Down arrow keys, and which stellations will be counted when you do an
enumeration ("Stellation→Enumerate...").
For a description of the different stellation criteria, see my paper
"Stella: Polyhedron Navigator".

To create a new stellation manually, you need to select and deselect
cells. Cells are the smallest volumes enclosed by sets of planes in
which the original model's faces lie, and are the 3D building blocks of
stellations. Actually, all cells of a certain type are usually referred to as
a single cell, and together they maintain the same rotational symmetry group as
the original model being stellated. A stellation is made up of some set
of all the possible cells for a given set of planes. There are a number of
ways to select/deselect cells, that is, to include them in or exclude them from
the current stellation.

Cell Diagram. Use Shift+Left-click to select
cells in the cell diagram view. Selected cells have a white border around
them. The cell diagram is an abstract representation of the relationship
between cells, and is usually not an intuitive way to create stellations.

Stellation Diagram. The stellation diagram shows the plane
from one of the original model's faces, with lines representing
intersections with all the other planes. Between the lines are
elementary regions, which are the tops and/or bottoms of stellation
cells. Use Shift+Left-click to select the cell below an
elementary region, or Shift+Right-click to select the
cell above (the mouse tips in the bottom right corner won't let you forget
how to use the mouse). This method is more intuitive than using the cell
diagram.

3D Stellation Diagram. The stellation diagram can also be
viewed in various 3D views (base polyhedron, dual polyhedra, base
stellation, and dual stellation). This is the most intuitive way to create
your own stellations, since the diagram is attached to an appropriate place
within the stellation itself. First select a face of the original model by
using Shift+Left-click on it. There are two options for
how the selected face should be displayed. By default it is highlighted in
bright white, and shows half-transparently through other faces. The other
option is to show the stellation diagram attached to the selected face
instead. Use the items on the
Selection→Selected Face Display submenu to choose. There
are also two buttons for these options on the options toolbar. Once the
stellation diagram is displayed, you can use the mouse as before to
select/deselect cells. Note that now Shift+Left-click
does two things. You can either click on regions of the stellation diagram
to select/deselect cells, or you can click on a different face of the model
to select a different face, and thus view its stellation diagram instead.

In any situation where you can select stellation cells using
Shift+Left-click, you can also use the following.

Ctrl+Left-click to select/deselect a cell along with
all its supporting cells, recursively back to the stellation's core. When
selecting cells in this way, the recursion also stops if it finds an
already selected cell. When deselecting cells, the recursion also stops
when it gets to a cell that is still needed to support some other selected
cell. So you can deselect a whole "peak" without creating a hole right
through to the centre of the model.

Ctrl+Right-click to select/deselect a whole layer of
cells, by clicking on one of the cells in that layer.

You can also stellate using an arbitrary set of facial planes, rather than
starting with the faces of a given polyhedron. This is more maths-intensive
for the user though, as you need to know the plane equations required. Use
"Stellation→Stellation Planes". A dialog box opens with various
options, initialised with values taken from the current polyhedron. At the top
you can select the symmetry group for the stellation, and below that you can
enter as many plane equations as you wish. Each plane will be repeated over
the symmetry group, so you only need to enter an equation for one of
each type of plane. The X, Y and Z fields represent a
normal to the plane (a unit vector perpendicular to the plane). This
vector will be normalised if it is not already unit length. The final
parameter to define each plane is Radius, which gives the distance of
the plane from the centre of symmetry. It may be zero to define a plane that
passes through the centre. Each field may be given as a decimal value, or an
equation like "4 + 2r3", where 2r3 means 2
times the square root of 3. You may also use tau or g as
shorthand for the golden ratio. It is very important that the values are as
accurate as possible, so use an exact equation where possible. If entering a
decimal value, I recommend 17 or 18 significant figures.

In the above dialog box, you may define a stellation plane using the
Add Measurement Plane button. No need to deal with numbers in this case.
Just define the plane you want first using Measurement
Mode.

This menu item selects any unselected cells that are totally hidden from
external view. For example, when stellating the dodecahedron, you might have
the cells selected to create the small stellated dodecahedron. Now whether the
central dodecahedron cell is selected or not makes no difference to how the
model looks from outside. Generally you want all those hidden cells to be
selected; otherwise they will lead to extra nets, for internal parts that won't
end up being visible!

Having all parts accessible is also one of Miller's Rules which were
used for The 59 Icosahedra.

Once you've made some complicated stellation from various cells, it's often a
good idea to fill in the inaccessible cells, since you probably don't want
hidden "bubbles" inside your model.

When using this feature, you will not see any difference in the model, but
you can see what's going on in the stellation diagram view (if there are
any inaccessible parts to fill in of course). You will also see a change in
the 2D or 3D nets view.

Exactly one of the items on this submenu may be selected at any one time.
The setting only affects "double", "coincident" or "pinched" edges in models,
i.e. where two solid pieces touch at an edge, e.g. when two faces sharing a
vertex intersect each other. From outside you can see the edge from either
side. When building a model, four parts in the nets meet at each such edge.
You might want to attach these in one of the following ways:

Tongue-in-groove (as named in Magnus Wenninger's
Polyhedron Models). Two faces are joined with their tabs pointing
out instead of in, then this is slid between the other two faces and glued
between them. In this case you need double tabs on both sides, i.e. faces
must not be joined along those edges in the nets.

Internal support. Here only one side must have double tabs,
and then the double tab will be glued to the back of the other faces
meeting behind that edge. So the faces on one side may be connected in the
net along the coincident edge.

No internal support. No tabs are required on either side of
the edge (i.e. faces may be connected along these edges in the nets).

Disconnected. This one's not generally of use in making a
model, but might help for creating nets of individual stellation cells. It
changes which faces connect to which at the edge. Instead of connecting
faces which appear to be connected from outside the model, it connects
faces which appear to be connected from inside. This means the solid
pieces will not be connected at all along those edges! Try the small
stellated dodecahedron with the central dodecahedron cell missing to see
the difference for this one. Compare the results in the unfolding net
view.

These settings generally make subtle differences in the nets produced.

Symmetries are transformations which leave a model looking like it
hasn't moved. They mostly come in two types: rotational and reflective. Use
"Display→Show Symmetry Axes" to display the rotational symmetry
axes (keyboard shortcut: S, also on options toolbar). In 3D views these
are indicated by an axis through the model with a small disc at either end.
The number of spokes in the disc indicates the order of rotational symmetry.
The different types of rotational symmetry axis are shown in different colors.
In the stellation diagram view these are indicated as points where the axes
intersect with the face plane, using the same colors.

Use "Display→Show Reflection Planes" to display the reflective
symmetries (keyboard shortcut: R, also on options toolbar). In 3D views these are
represented by great circles around the model in the reflection plane.
In the stellation diagram view these are shown as dashed lines where the
reflection planes intersect with the face plane.

The symmetry group of a polyhedron is the collection of all its
symmetries. The options toolbar has a field that shows the rotational
symmetry group (e.g. "Icosahedral"), and another field showing the reflective
symmetry group within that (e.g. "Horizontal Reflection"). To the left of
these is another field showing two common symbols used for the current symmetry
group: Schönflies notation, Orbifold notation, and Coxeter notation.

Another good web page showing the different symmetry groups
is
http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/Solids.html.
The only groups missing from this site are icosahedral symmetry, both with and
without reflections. The h subscripts on that page correspond to
"Horizontal Reflection" in Stella, and the d subscripts
correspond to "Diagonal Reflection". Notice that the rotational symmetries of
a tetrahedron can be combined with either of these reflection types
(Th and Td). The latter is the symmetry
group of the regular tetrahedron, while the former is known as
pyritohedral symmetry.

Some symmetry groups are subsets of others. All three symmetry group fields
in the toolbar have drop-down lists, which let you see all the sub-symmetry
groups available, and choose one. Changing to a sub-symmetry group will affect
stellation, faceting,
augmentation, and coloring of the model. Note: you
can use "Options→Recolor Sub-Symmetries" to prevent the model's
color from changing when choosing a sub-symmetry group.

Some symmetry groups contain rotation-reflection symmetries, made from a
combination of a rotation and a reflection where neither step represents a
symmetry of the group on its own. These appear as "Rotation-Reflection" in the
reflection symmetry drop-down list, and only occur in conjunction with some
dihedral symmetry groups. When reflection symmetries are displayed, these are
represented as a wavy circle around the model, rather than flat, with waves
matching the rotation required. A special case of this is known as central
inversion, where the only symmetry in the group is a reflection through a
point at the centre of the model. This is the same as a rotation-reflection
where the rotation is 180 degrees. This special symmetry group is represented
by a few double-headed arrows through the centre of the model. Your homework
is to try to create polyhedra in Stella4D that show these special
symmetries!

Items in this submenu allow you to enter a different mode, where the mouse
behaves differently from normal. Exactly one of the items will be selected at
any one time. The first item on the list is the default mode, so to exit one
of the other modes, either select this item, or just hit
Esc. There is also a toolbar containing a button
for each of these modes.

You can tell when you're in a different mode from usual by holding down
Shift and seeing what shape the mouse pointer becomes. Navigation
with the mouse acts the same in all modes. The difference is what happens when
you're holding down Shift and/or Ctrl.

If you don't like the net created, you can force the program to cut
certain edges. This makes it rebuild the nets, connecting different
faces together instead.

Shift+Left-click: Cut/uncut edges of selected type.

Shift+Right-click: Cut/uncut a single edge.

Uncutting an edge does not guarantee the faces will be attached in the net,
just stops them from being forcibly cut.

Note that cutting one edge may cause other faces to join together in a net
where they were previously not connected. This is because the nets are rebuilt
from scratch. So you can use cutting as a way of forcing other faces to join
up, but only indirectly. If you didn't want those faces to connect, then you
can always cut their shared edge too.

As an example, use Shift+Left-click on an edge of the icosahedron.
All edges are the same type, so they are all cut, so you end up with only one
face in each net. Use Shift+Right-click instead to cut just one edge.

Measurement Mode lets you measure the distance and angle between any two items,
where each item may be a
hyperplane (for 4D polytopes),
plane, line or point. The distance and angle are shown on the screen, and the
items highlighted in light blue, along with some construction lines to help
clarify what's being measured. Items may be defined via the mouse:

Shift+Left-click: Select a new item.

Shift+Right-click: Extend current item.

Ctrl+Left-click: Replace most recent item with selected
one.

Ctrl+Right-click: Extend current item.

So use Shift+Left-click to select a vertex, edge or face, and then
again to select another and see the distance and angle between them. If you
want to specify an item that doesn't correspond to an existing entity, such as
a plane through three arbitrary vertices, then you can build up this plane by
specifying each vertex in turn. Use Shift+Left-click for the first
one, then use Shift+Right-click to add the others. You can even
specify a plane through an edge and a vertex by
Shift+Left-clicking the first and
Shift+Right-clicking the second.

Normally at most two items are remembered at a time, and the distance and
angle measured between them. So when you specify a new item, the oldest item
of the previous two is discarded. You can use Ctrl instead of
Shift with the above mouse operations if you want to replace the most
recent item rather than the oldest. This retains the earlier item, so is handy
if you want to fan out measurements from the same point repeatedly.

These is one occasion where three items are remembered rather than two.
This is when you select three vertices in a row. The distance between the last
two is shown, along with the angle subtended at the middle point.

You may also specific a point at the centre of the model, even when there's
no vertex there to click on. There's a button which appears on each view's
title bar when in Measurement Mode to achieve this. It shows a circle with a
dot in the middle. This allows you to measure distances from the centre of the
model. Or you can use it as the first point in a line or plane through other
arbitrary vertices or edges by using Shift+Right-click as above.

In addition to appearing on-screen, the distance and angle are shown in the
status bar at the bottom of the window. You can also access the currently
measured distance via "Scale→Measured Distance". This also
lets you change the distance, which affects the scale of
the whole model, as used for printing nets and some other operations.

Note, in the base or dual view, you may only click on true vertices of the
model, not virtual ones caused by intersecting faces, unless you turn on
display of stellation vertices
(see Showing Vertices and Edges).
You may also click on points in the stellation diagram.

Aside from measuring things, Measurement Mode has other uses.
The most recently defined item may be used in the following ways:

To define the cross-section direction and depth. Use
"Section→Cross-Section Direction→Measured Item First". See
Cross-Sections.

This mode lets you easily copy colors, images, or "hiddenness" between
faces.

Ctrl+Left-click: copy attributes from the previously
selected face to the clicked face.

Shift+Left-click: copy attributes from the previously
selected face to all faces of the same type as the clicked face (use
"Options→Maintain Reflexibility" to choose whether reflected faces
should also be changed).

In either case, you may hold the left mouse button down and drag the mouse
across multiple faces to apply the color/image to all of them, like painting.
No need to click on each one.

If the selected face was hidden, then the "hiddenness" is copied, that is,
other faces may be easily hidden too by Ctrl+Left-clicking on
them. In this case the color/image of the face is not changed (it is presumed
that it is only the hiddenness that you are interested in copying).

Ctrl+Right-click and Shift+Right-click:
select a face without changing its color. This lets you choose a new face to
start copying attributes from. In this mode, the face is only highlighted
white briefly when you select it, and then returns to its normal color, since
it is important to see true face colors in this mode.

If you plan to do a lot of augmentation (see
below), this mode can be useful. One of
the options when augmenting is to only augment faces of the same color. So
select one of the faces you want to augment, then copy that color to all the
other faces you want to augment. This lets you quickly and easily select a
collection of faces for augmentation.

If the selected face does not already have an image on it when entering this
mode, a file browser will open allowing you to select one. This is the same as
if you had used "Image→Load Image" (see
Images above). You may cancel the file browser if you
wish to use Image mode without loading any new images.

Image mode lets you accurately position images on faces however you want.

Shift+Left-drag: move an image on a face

Shift+Right-drag: scale an image on a face

Shift+Left+Right-drag: rotate an image on a face

Ctrl+Left-click: select face without affecting image

Ctrl+Right-drag: stretch image

Ctrl+Shift+Left-click: swap image with previously selected
face

Compare these three operations with navigation in a 2D view. Notice the
similarity when you don't hold down Shift? This makes it easier to
remember which combination of buttons does what.

Use Ctrl+Left-click to select a face without affecting the
image on it. And use Ctrl+Right-drag to stretch an image on
a face (changes the aspect ratio of the image). Note: when using
spherical or cylindrical mapping, the image must always wrap all the way
around, so scaling and stretching the image both have the same effect, to scale
it vertically, but leave it wrapped all the way around horizontally.

Use Ctrl+Shift+Left-click to swap images between
two faces. The images are swapped between the face you click on, and the face
that was previously selected. This is useful in conjunction with
"Image→Load Image List...". You can load a bunch of images at
once, then swap them between faces to get the arrangement you want.

In this mode, you can Shift+Left-click on a series of
vertices of a model to create a new polygon. A helpful message will appear
when you enter this mode, explaining what operations are available via the
keyboard or mouse. You will see the new polygon being drawn as you go (it will
show half-transparently through other faces). The points must be coplanar, and
once three or more points have been added, other coplanar points are
highlighted in green. If you wish to add a point at the centre of the model,
use the button showing a circle with a dot in the middle at the top of the
view. When finished, hit Enter or Shift+Right-click anywhere
to accept the polygon (or hit Delete or
Esc to cancel it).
You can also accept or reject the new polygon with the tick and cross buttons
at the top of the view.

Once accepted, the new polygon will appear as another net, on its own (hit
PageDown in the Unfolded Net view a few times to find it), and you can
print it out. I use this to create supporting pieces to glue inside models for
added strength.
See my page about the great dodecicosidodecahedron for an example:
http://www.software3d.com/GreatDodecicosidodeca.php.
You may define more than one such polygon. Creating new polygons in this way
is also the first step to creating faceted polyhedra
(see next section).

Some other useful controls in this mode are as follows:

Backspace: Remove the last vertex added to a facet. This
is useful when you don't want to delete the whole facet and start from scratch.

Ctrl+Right-click: Create a complete facet from an
existing face by clicking on it this way.

"." (on the keyboard) or the "..." button (on view's title bar):
Auto-complete a facet. Yes, the program can often figure out how to complete a
facet for you! You have to enter at least three vertices of the facet
yourself, or sometimes more if the repeated sequence is longer, then hit "."
and the facet will be finished for you. Only the first three vertices are
required for any regular polygon or polygon with equal angles and alternating
edge lengths (such as rectangles).

A faceting of a polyhedron is one which shares the same vertices, but
has different faces connecting them. Faceting is actually the dual process to
stellation, the former keeps the vertices while the latter keeps the facial
planes, but I won't go into that here!

The previous section described how to make extra polygons, or facets,
using the original vertices of a model.
When one or more of these facets have been created, you may use
"Faceting→Create Faceted Polyhedron" (keyboard shortcut:
Ctrl+Shift+F) to create a new polyhedron from them. Each facet is repeated
over the current symmetry group, and the resulting faces are used as the faces
of a new polyhedron, which becomes the new base model (so you can see its nets,
dual etc). Not all combinations of facets you might create will lead to a
valid polyhedron however, in fact most won't. So you need to be careful when
creating the facets. An even number of faces must meet at every edge (and
usually you will want exactly two faces).

The Preview Faceting view lets you see how your faceted model is
coming along before you've finished it. To use this view, select a spare view
whose type you don't mind changing (by clicking in it) and use
"View→View Models→Preview Faceting"
(or the toolbar equivalent). Create a facet (using one of
the other 3D views and Faceting Mode) and
you will see how it is repeated over the symmetry group. Edges with only one
face are highlighted in green. These are edges
where you need to create another facet to share that edge.
Orange edges have an odd number of faces, but
at least three. Here you need to either delete one of the facets, or add yet
another one. Red highlights edges where an even
number of faces meet, but greater than two. These are acceptable, but often
undesirable. Finally, if exactly two faces meet at an edge, but they are
coplanar, then purple highlighting is used.
This is also often undesirable, but not disallowed. Edges where exactly two
non-coplanar faces meet are not highlighted at all. Generally you are trying
to get rid of all highlighted edges. When you have, create the model with
Ctrl+Shift+F, or just click on the left-and-down arrow at the top of
the Preview Faceting view.

Use Ctrl+Left-click in the Preview Faceting view to select
a facet. Then you can click the "X" button on the title bar to delete that
facet. You can also find it in the Unfolded Net view (with PageDown)
and delete it using the "X" in that view's title bar.

Since facets are repeated over the symmetry group, you can use the
symmetry drop-down lists (see Symmetries) to
select a different symmetry group, and the facets will be repeated accordingly
in the Preview Faceting view. You can also choose whether they should be
repeated over reflection symmetries with "Options→Maintain
Reflexibility".

It can be hard to find sets of facets that go together correctly to create
valid polyhedra. Luckily Stella provides a way to automatically step
through all possible valid facetings, and also to see what individual facets
are possible. With the Preview Faceting view selected, simply hit the
Up arrow key (or the green up-arrow button with the "F" next to it) to
step through all the possible facetings.
The Faceting menu also plays a part. It contains the following items:

Create Faceted Polyhedron. (keyboard shortcut: Ctrl+Shift+F)
As described in the Faceting section, this creates a
new base polyhedron from the current set of facets.

Up/Down Arrows in Faceting View. Submenu for deciding what the
Up and Down arrow keys should do when a Preview Faceting view
is selected. Options are:

Step Through Facetings. The Up arrow will step through
each possible complete faceting, creating whatever facets are required.
The Down arrow should step through them in reverse order, though
this is currently only implemented for Isohedral facetings.

Step Through Edges. Use Up and Down arrow keys
to step through each different type of edge that a faceting might use.

Step Through Planes. Use Up and Down arrow
keys to step through each different type of facial plane that a faceting
might use.

Step Through Facets. Use Up and Down arrow
keys to step through each different type of facet that a faceting might
use.

Faceting Criteria. This submenu lets you choose which kind of
facetings you're interested in. When you step through the facetings, only ones
matching this criteria will be included. The options are in order from most to
least restrictive.

Isohedral. Only accept facetings where every face is the same
type. This generally greatly reduces the number of facetings available to
a small and interesting set.

Tidy Facetings & Duals. Accept all facetings which are
tidy and whose duals are also tidy. Tidy simply
refers to the more conventional type of polyhedron (or compound), which is
finite, has exactly two faces meeting at each edge, each face visits a
vertex at most once, and coplanar faces may not share any vertices.

Tidy Facetings. Similar to above, but accept facetings where the
dual is not tidy. The most common difference is that we may now include
facetings with faces in planes that pass through the centre of the
polyhedron. The duals of such polyhedra are always infinite.

Allow Coplanar Faces Sharing Vertices. Similar to above, but we
relax the requirement that coplanar faces mustn't share a vertex.

Minimum Facet Types. Only accept facetings with at least this many
face types (normally set to 1).

Maximum Facet Types. Only accept facetings with at most this many
face types. Enter 0 to indicate no limit (normally set to 0).

Accept Partial Facetings (Not Using All Vertices). Allow facetings
which don't make use of all vertices. Symmetry is still maintained, so this
won't make a difference for polyhedra with only one type of vertex, but when
there's more than one type, some may be left out.

Only Accept Spiky Facetings. Only accept facetings where no edges
lie on the convex hull. This ensures only spiky polyhedra.

Enumerate Facetings. Count the number of facetings using the current
criteria.

Faceting Diagram Options. This submenu is explained in the
Faceting Diagram section below.

Delete All Facets. Convenient way to delete all the facets and start
again.

Another related view type is the Faceting Diagram
("View→View Diagrams→Faceting Diagram" or the toolbar
equivalent). This shows the faceting diagram (which is reciprocal to the
stellation diagram for the dual). There is a different faceting diagram for
each different type of vertex of a polyhedron, so try selecting different
vertices in one of the 3D views (by double Right-clicking on them) to
see the faceting diagram for that vertex, or use the yellow left and right
arrows on the title bar to cycle through them.

The diagram has the following features:

Each white point in the diagram represents one of the other vertices of the
model, all projected back towards the main vertex and onto the plane of the
diagram. Use double Right-click or Shift+Right-click to
select a vertex from the diagram.

The big blue point in the middle represents the centre of the model.

Each line represents a possible face in a faceted version of the current
polyhedron.

Lines are colored according to how far from the centre they pass (yellow
near the centre, cyan for the furthest out).

If a facet has been created, then it is shown as a heavier line, and
colored to match its color in the Preview Faceting view. You may select these
facets with Shift+Left-click.

For an uncluttered view, see the "Faceting→Faceting Diagram
Options" submenu. It contains "Hide Diagram when Facets Exist"
and "Hide Vertices when Facets Exist".

Stella4D has nine memories. These are like the memory on a
pocket calculator, except instead of storing a number, they store a polyhedron.
Quickest way to use memories is to hit m followed by a number from 1 to 9 to
store a model in memory, and hit Shift+M followed by a number to
retrieve a model from memory. The number indicates which memory slot to use.

You can also deal with memories via the menu:

Edit→Put Model in Memory submenu: put the model from the
current view into one of the nine memories. Note: the model put in memory is
the one you see in the current view, which may be the base model, its dual, a
compound of the two, or a stellation of either one. You may not put infinite
dual models in memory, nor stellated models with faces that have holes in them.

Edit→Retrieve from Memory submenu: retrieve a model from
one of the nine memories. This then becomes the base model. Note: if you had
put a dual model in the memory, then this is now the base model.

Edit→Swap with Memory submenu: swaps the model in the
current view with the model in one of the nine memories. The model that was in
memory then becomes the new base model.

Edit→Clear Memory submenu: empties one or all of the nine
memories.

Edit→Add/Blend from Memory submenu: adds
the model in one of the nine memories to the model in the current view. The
result becomes the new base model, which will now be a compound of the two
models. If you want the relative sizes of the two models to be different, then
change the scale of the model before using "Add/Blend from Memory",
but after putting the other model in memory first. Scale can be changed in
various ways via the "Scale" menu. If the
resulting compound has coincident faces, an offer is made to remove these in
pairs. If the model has coplanar faces sharing an edge, an offer is made to
blend them together. In either case, the result may now no longer be a
compound, with the two parts now sharing geometry.

Use "Poly→Augment Polyhedron..." (keyboard shortcut: A) to
augment, excavate, or drill the current polyhedron. These
are all basically different aspects of the same function. Augmentation
is simply attaching one polyhedron to another at some common face.
Excavation is where one model is subtracted from another at some common
face, leaving a dent. In either case, any faces that coincide exactly between
the two models are removed. The face where the two models were attached always
falls into this category, but others may too. When this happens during
excavation it is called drilling, as the result is a hole right through
the model.

When choosing to augment a model, a dialog box appears with the following
sections.

Augment using... Choose what model you want to augment the existing
polyhedron with. You can use any of the following: pyramid, cupola, prism,
antiprism, a model in one of the nine memories, any built-in model, or a model
from a .stel file. The model you choose may have more than one type
of face matching the face you selected on the original model. If the model is
from memory or a file, then the default face used will be the last one selected
before it was saved. You can also select a different face during preview (see
below), but it can save you some time if you think ahead!

Which faces to augment? Choose whether to augment just the
selected face, or all faces of the same type or shape. You can tick
"Only faces of same color", then even if you've chosen to augment all faces of
the same shape, only those of the same color as the selected face will be
augmented. This can be useful in conjunction with Color Mode. Paint the faces you want to augment in a new
color, then augment only those ones.

Direction Choose to augment or excavate.

Coloring Choose from the following options:

Re-color faces according to current color scheme. The entire
model will be re-colored, ignoring any existing colors.

Keep colors from original models. Colors are maintained from
both the original model, and the model being added. Note: if augmenting
using a pyramid or other model with no predefined colors (i.e. not from
memory, file, or built-in), then the color is taken from the selected face
of the original model.

Inherit colors from augmented faces. Models being added are
each colored entirely to match the face they are being added to.

Use a single color for new faces. Click the "Color" button
besides this option to select a single color that will be used for all the
added models. Faces of the original polyhedron remain unchanged.

Use any single new color for new faces. Stella will
choose a single new color (one not already appearing in the model) and use
it on all the new faces. Faces of the original polyhedron remain
unchanged. This is convenient when performing a series of augmentations if
you want a different color at each step.

When to preview By default an augmentation preview is always
presented (see below), but you may choose to only show the preview when there
is more than one orientation to choose from, and complete the augmentation
immediately otherwise.

Preview: Wire-frame, Solid or Both Choose how the preview should be
displayed.

Keep coincident faces Override the default behaviour of removing
coincident faces between the original model and augmented parts.

When augmenting with pyramids, the pyramid height may be selected with the
following options:

Guess. Same as "Keep edges same length" when possible,
otherwise same as "Face radius".

Keep edges same length. Use a height for the pyramid that ensures
all edges of the pyramid are the same length. If you are trying to maintain
regular faces, you'll want to use this option.

Face radius. Set the height of the pyramid to the radius of the
face.

Face diameter. Set the height of the pyramid to the diameter of the
face.

Touch model radius. Set the pyramid height such that its apex just
touches the circumsphere of the original model. When augmenting, it touches
the sphere on the same side of the centre, when excavating it touches the
sphere on the far side. This option can be handy if you want to make a spiky
model and ensure that all spikes reach the same height.

New edge lengths = 1. Set the length of edges reaching the apex of
the pyramid to 1. This is useful in conjunction with the Scale setting
below, allowing you to set the exact length of the new edges.

Height = 1. Set the pyramid height to 1. This is useful in
conjunction with the Scale setting below, allowing you to set the exact
height of the pyramid.

When augmenting with prisms, the prism height may be selected with the
following options:

Average edge length. Prism height is set to the average length
of edges around the face being augmented.

Shortest edge length. Prism height is set to the shortest edge
length around the face being augmented.

Longest edge length. Prism height is set to the longest edge
length around the face being augmented.

Height = 1. Set the prism height to 1. This is useful in conjunction
with the Scale setting below, allowing you to set the exact height of
the prism.

The Scale setting can be used to scale the height of augmentations when
using pyramids, prisms, antiprisms or cupolae. In most cases it scales the
height of the augmentation directly, but there are some useful exceptions when
augmenting with pyramids. For the "Keep edges same length" and "New
edge lengths = 1" options, the scale applies to the edge length used
instead of the height. With "Touch model radius", the scale applies to
the model radius, then the pyramid height is chosen to touch that new radius.

After clicking OK, an augmentation preview mode will be entered.
Here a preview of the result is shown. Depending on your choices above, it
will normally show the augmented parts as solid with a wire-frame outline (the
wire-frame shows through other faces too, for easy identification).
A message at the top indicates how many orientations are possible for the
augmentation.

The following controls are available during the augmentation preview.

Left and Right arrow keys, or the yellow
left and right arrow buttons at the top of the view:
step through all possible orientations for the model being added while keeping
it connected using the same pair of faces.

Up and Down arrow keys, or the yellow
up and down arrow buttons at the top of the view:
if the model being added may be attached at more than one type of face, then
step through each possibility.

Ctrl+Left-drag: interactively change the height of the
augmentation when using a pyramid, cupola, prism, or antiprism.

Ctrl+Right-drag: change the radius of the top of the model
being added when using a cupola, prism, or antiprism.

Ctrl+Shift+Left-drag: introduce a gap between the original
model and the model being added. Drag the mouse to adjust the size of this
gap.

Ctrl+Shift+Right-click: close the gap between the original
model and the model being added.

A key: hit "a" again to reopen the augmentation dialog box
and change options.

Enter key or tick button: accept the augmentation
and create the finished model.

Augmentation parameters are remembered during a session, so if you augment
a lot of times using the same model, you will mostly just have to hit
"a" followed by Enter, without having to change anything in
the dialog box. This makes it very quick to build up a big structure.

Use "Poly→Put Models on Faces/Vertices" to put any model on the
faces or vertices of another model. This is similar to augmentation, but
rather than attaching the new model at a face common to the two parts, here the
new model is not attached to the original at all, but rather just sits in space
on its own. It may intersect with the original model, but there will generally
not be any edges connecting the two parts. It is more like a compound in this
regard.

A dialog box appears with various options. In the Source section you
may choose where to get the new model from (a memory slot, a built-in model or
dual, or from a .stel file).

Below this is the Placement section. Here you can choose whether to
put the new model on faces or vertices of the original, and at what
height above these features to put the model. At a height of zero, the
model will sit neatly on a face or vertex. Negative values will sink the model
into the face/vertex. At -100% the model will be completely buried inside the
other model (at least at the face or vertex where it is attached). Positive
values cause the new model to float above a face or vertex. You may also
choose whether to put a copy of the new model on just one face/vertex, or all
those of the same type, shape, or shape and size, or even on all faces/vertices
(regardless of shape).

Next is the Size section. Here you can choose how big the added
model should be. The size can be based on one of three options:

Fit to current face/vertex. The model is scaled to just fit inside
the face/vertex to which it is added. For faces that means the added model's
radius is set to match the inradius of the face. For vertices, the model's
radius is set to match half the distance to the nearest neighbouring vertex.

Fit to smallest face/vertex of part. This is similar to the above
method, but rather than fitting the model to the face/vertex on which it is
placed we instead fit it to the smallest face/vertex of the same part (e.g. the
same component in a compound). With the previous method, models added to all
the faces of a truncated cube would end up two different sizes. Big ones on
the octagons and small ones on the triangles. This method ensures they all
come out the same size (all small enough to fit in a triangle). The reason for
fitting in the smallest face of one part rather than the whole model is that
once one model has been added, it brings with it new smaller faces. If you
decide to add another model, to another face of the original base model, you
don't want it to be scaled smaller than the first in order to fit in those even
smaller faces.

Same size as base part. Here the added model is simply the same
size as the model it is being added to. In conjunction with the Scale
setting (see below) this gives you precise control over the size of the model,
especially if you know the ratio you want between the two.

The above choices are good starting points, but the Scale setting then
has the final say. The value you input here is multiplied by the size
determined above to get the final size of the added model. For example, set
the scale to 0.5 to make the added model half the size it would
otherwise be, or 2 for double the size.

The Color section lets you choose how to color the added model.
It can use its original colors (e.g. whatever colors it had when you put it in
memory), or it can use a single color of your choice. Click on the colored
button to select a color.

Finally, there's one more option at the bottom of this dialog bow,
Retain original model centre. If this is ticked, then the resulting
model will continue to rotate on-screen around the same point it previously
rotated around. Otherwise a new model centre is calculated as the centre of
mass of all vertices. This may seem off-centre, as it will be dragged closer
to the added model, due to the new vertices from that model.

[Not available in demo version]Stella4D includes a library of over 400 additional models, aside from
the many that are built-in. The models in this library were all made using
features of the program. They all exist as files on your computer, with the
".stel" file extension, and are located in a folder called
StellaLib, under the folder where the program was installed. Feel free
to add your own ".stel" files under this folder, and even your own new
sub-folders. They will show up in the software.

To select a model from the library you can select the Stella Library
option from the category drop-down list on the main toolbar. The
sub-categories will then show up in the model drop-down list beside it. Select
a sub-category to see the models in that sub-category in the model list, and
select one of those to open it.

Alternatively, use "File→Polyhedron List...", hit
Ctrl+N, or click the matching toolbar button to open the list of
built-in polyhedra. A window opens with a list of all built-in models,
including the models in the polyhedron library. If Stella Library in
the category list on the left has a "+" beside it, click it to open the
list of sub-categories. Select a sub-category to see the list of models in
that category on the right, then double-click a model to open it.

Files in the library are organised into the following sub-categories:

Augmented Uniforms
Uniform polyhedra that have been augmented with pyramids or cupolae.

Brückner
In 1906 Max Brückner made some amazing paper models, all of which were
isogonal (all vertices the same) and isohedral (all faces the same). The
polyhedra were all self-intersecting and mostly quite complex. An amazing
achievement in 1906! Now most of these models are available in this category
(all except for the ones with dihedral symmetry).

Compounds
Compounds of multiple polyhedra. Note that the dual compounds are also
available. So to see the compound of five octahedra, load the compound of five
cubes and look at its dual.

Facetings
Various polyhedra created using faceting.

Geodesic Hemispheres
A collection of geodesic domes.

Geodesic Spheres
A collection of geodesic spheres, created using Great Stella's
"Poly→Create Geodesic Sphere" operation.

Leonardo-style
Some Leonardo-style models, that is, models with solid edges but hollow
faces.

More Stewart Toroids
More Stewart Toroids, to add to the built-in list. These regular-faced toroids
are mostly from Stewart's book "Adventures Among the Toroids", or extensions of
that work.

Parts
Polyhedra which can be constructed more robustly using multiple parts. You
build one part, then the other, and glue them together.

Rectangular Isohedra
The four polyhedra with rectangular faces which are all of the same type.

Stellations
Polyhedra created using stellation.

Sub-symmetric
Polyhedra created using sub-symmetric stellation.

Topological
Models where faces have been partly cut away to reveal the internal structure,
where the parts remaining only touch at true edges and vertices of the original
polyhedron (avoiding false edges where faces may have intersected though not
sharing an edge).

You will also find the Geomag Library category, which contains fun
models that can be constructed with Geomag's construction
kits.

If all else fails, and you can't find a way to create your favourite model in
Stella4D, fear not! You can always import it from an external file.
Currently the formats supported for import are OFF and DXF. OFF is preferred
and more likely to work than DXF. It's a simple format and you should be able
to figure it out from the example below. Please be sure to use as much
precision in the vertex positions as possible, or you may run into problems. I
recommend using 17 decimal places if you are creating files in this format.
The OFF models must also be properly closed surfaces (with an even number of
faces meeting at each edge). If not, it can't be opened, but you will be
offered the choice to create the convex hull of the model in the OFF file,
which allows you to get at least some of the vertices in. You may then wish to
use Faceting Mode to connect these vertices together
in other ways. You could even create OFF files with only vertices (no faces)
and let Stella4D figure out how to connect them together for you.

To import an OFF file, simply use "File→Open" as you would to
open a .stel file. You will find that the file browser also shows
.off files. Select one in the browser to see some information about
it at the bottom of the browser, including number of faces, edges and vertices.
Note: number of edges may be incorrect, as the format does not require this to
be provided correctly in the file. It will probably either be correct, or
zero.

3D DXF files may be imported in a similar way, but I fear they often lack
the accuracy required for Stella4D.

Here's an example OFF file. Copy the text into a file with a name ending
with ".off" in order to open it in Stella4D.

Lines starting with "#" are comments and are ignored upon import.
The number of edges is also ignored and may just always be given as zero.

There is a 4D version of the OFF file format which can be imported too, and
Stella4D also accepts an extended version of this format which specifies
cells in addition to faces. The cell section is optional, but
Stella4D may have problems figuring out how to put faces together to form
cells in some cases without it, and there may be more than one way to do this.

Below is a sample 4D OFF file. Copy the text into a file with a name ending
in ".off" in order to open it in Stella4D.

As with the standard 3D OFF format, the number of edges is also ignored and
may just always be given as zero. Colors appear after each cell rather than
each face (colors given after faces will be ignored). Each cell is given as
the number of faces followed by the index for each face around the cell. As
with 3D, you may provide only vertices (with no faces or cells) and
Stella4D will create a 4D convex hull for you.

3D models can be exported in a variety of formats. This is achieved by using
items from the "File→3D Export" submenu. Generally the model is
exported from whichever view you have selected (e.g. you might export a dual,
or a stellation, or even a model morphed between two duals). Available formats
are DXF, POV-Ray, VRML, OBJ, OFF and STL. The dimensions of the model exported
can be controlled using the Scale menu.

A dialog box provides various options, which vary a bit between formats.
Options include:

Select an orientation for the model:

Use current orientation

Use default orientation

Look down or sit on a symmetry axis

Look down or sit on a selected face, edge or vertex

Sphere at vertices and/or cylinders along edges may be exported optionally
as well. For most formats these are approximated using additional geometry.
With VRML you have the option of using VRML's native support for spheres and
cylinders.

Note
also that the exported model will be oriented the same as it is on screen.
This gives you powerful control over the orientation of the exported model, but
can also lead to confusion if you didn't want the model to have a random
orientation. I suggest you place the model in a nice orientation before
exporting, by using one of the items from the "View→Orientation"
submenu.
Note: 4D models can only be exported as their 3D projections or other 3D
aspects, not as raw four dimensional data. You can however import 4D models,
see above.

2D images and videos can be exported in various formats. To do this, use the
"File→Export Image/Video..." menu item
(keyboard shortcut: X). An image or video of the current view is
exported (make sure you click in the view you want to export first). Videos
may include animation of various things including rotation,
4D rotation,
net folding, dual morphing, and
cross-section depth changing. You may also export
tours as videos. Videos may be previewed in the main
window.

A dialog box appears with various options. This dialog supports some
descriptive tool-tips, so if you get lost, move the mouse over a control to get
additional advice. Options in this dialog are as follows:

Folder. The destination folder. A history of previous folders used
is kept for quick access. Click on the down-arrow to the right of the folder
name to open a drop-down list of previous folders. Select one to use that
folder.

Save as. The file name to use (without a path). To the right of
this is a drop-down box showing the type of image or video to be saved. Select
the type you want. AVI is always video. GIF may be either an image or an
animated GIF. All other formats may be either an image, or a video represented
as a sequence of individual images.

Browse. Browse for the destination folder and file name.

<Common resolutions>. Drop-down list containing a collection
of common video resolutions. Selecting an option fills in some of the fields
in the rest of the dialog box, including the pixel width and height, pixel
aspect ratio, and frame rate. Normally the pixel ratio should be "1", but
standard definition PAL and NTSC have slightly rectangular pixels, so the
appropriate ratio is filled in for those cases.

Orientation. Select an orientation for the model, such as looking
down or sitting on one of the symmetry axes, or looking on or sitting on the
selected face/edge/vertex/symmetry axis.

Below this, options are divided into the following sections:

Resolution

Image size same as window. Tick this box to export an image the
same size as the current view. The following two items will be greyed out
if this item is ticked, but will show the size to be used.

DPI. Pixel density (Dots Per Inch). The
physical size and pixel density don't generally need to be set, unless the
image will be imported into a package like MS Word or sent to a publisher.

Pixel aspect ratio. You almost always want this to be "1"
meaning that your pixels are square. Only change it if you want non-square
pixels, such as standard definition PAL or NTSC, which defines pixels that
aren't quite square ("128/117" for PAL and "4320/4739" for NTSC).

Fit geometry to image

Fit tightly (each frame). When ticked, geometry will be resized
to fit perfectly in the image being saved (like hitting "t" in a view).
You generally don't want this for video because as a shape rotates, or
folds, or morphs, it will grow and shrink a little each frame to re-fit the
new geometry.

Fit with room for rotation etc. When ticked, geometry will be
resized to fit in the image being saved, but leaving room for possible
rotation or changing slicing depth etc. (like hitting "f" in a view). If
neither this nor the above option are ticked, the geometry will appear the
same size as in the view (but resized to the image dimensions).

Borders when fitting. Rather than fitting exactly with one of
the above two options, this allows you to specify a small border in order
to leave some space around the geometry.

Quality

Over-sampling. A higher value gives better
anti-aliasing, i.e. smoother edges and less pixelated. The lowest
value, 1, will produce an image exactly as it is on-screen. With
old computers you may run into problems using a high value (3 is the
highest value allowed), as the process may run out of graphics memory. If
anything looks wrong with the image, try a lower setting.

JPeg Quality. This setting is only available when you have
chosen to save a JGP image. At higher values (up to 100) the JPeg
image saved will be higher quality, but the file will be larger. At lower
values (down to 0) the file will be smaller, but the image quality
will be worse.

Animation options

Export animation. Tick this to export a video. Untick to
export a single image. When unticked, the other controls in this section
will be greyed out.

Export current tour. Tick to export the current
tour as a video. Various other controls will be
greyed out, as the tour will control all aspects of the animation.

Rotate around. Choose vertical axis to rotate the model
around an axis up the screen, or selected item to rotate around the
last selected item (face, edge, vertex, or symmetry axis). This is the
same as when pivoting around an axis normally (see
Navigation).

Number of rotations. Number of complete rotations around the
axis chosen above over the length of the animation. As with all fields
expecting a decimal number, you may use an
equation here. For example, if you are rotating
around a 5-fold symmetry axis, then the model will align with itself again
after just one fifth of a full rotation, so enter "1/5". This would be
enough for a continuous looping animation.

4D rotation. For 4D models, tick this to do 4D rotation instead
of 3D rotation. The 4D rotation will occur through the last selected item
(cell, face, edge or vertex).

Tilt forward. Angle in degrees to tilt model forward (rotated
around a horizontal axis on the screen). The rotation axis also tilts, to
remain in the same place relative to the model. When rotating around a
vertical axis, this tilts the model and axis forward, providing a more
pleasing view of the model.

Unfolding ratio / Morph ratio / Cross-section depth. One of
these three options may be available depending on the view you are
exporting, allowing you to animate net folding, dual-morphing, or
cross-section depth changing. You may enter the ratio at the start and end
of the animation. 0.0 represents one extreme in the transition, while 1.0
represents the other extreme. Since you may want your model to completely
unfold and then start folding up again, or morph from base to dual and then
start morphing back again, values between 1.0 and 2.0 act in reverse. So
to perform a complete transformation, from one extreme to the other and
back again, set the range from 0.0 to 2.0. Set the end ratio to higher
even numbers to repeat the whole transformation more times.

Frame rate. Number of frames per second for the video. This is
followed by a choice for interlacing. These days "Not interlaced" is
probably what people will want most. But if your destination video format
is interlaced, then choose another option, maybe doing a test video to
check which field order you require.

Length. Length of video in seconds, followed by length in
frames.

Seamless looping. Tick this if you intend your video to run
endlessly in a loop. This makes the animation hit the final frame just
after the end of the video, since this is where the first frame is
expected to come in again. Untick if you expect your video to run just
once. In this case the final frame of the animation is included as the
last frame of the video. When exporting a video of a
tour, there's one more difference. A seamless video
includes an extra transition from the final event in the tour back to the
first event again. When unticked, the video stops after the final event,
leaving out the transition that would take us back to the first event
again.

AVI video compression. This section is only relevant when exporting
an AVI video.

The first field is a drop-down containing a list of all the codecs
available for AVI compression on your computer. Select whichever one you
wish to use for your video.

About. Tells you about the selected codec.

Configure. Allows you to configure the selected codec (if the
codec provides this).

At the bottom of the dialog box are controls for previewing the export. When
Preview is ticked, a preview of the image or video to be exported is
shown in the main window. In the case of video, playback controls allow you to
play the animation forwards or backwards, fast forward, fast rewind, and step
one frame forwards or backwards. The preview takes the frame rate into account
too, so if you are exporting at 5 frames per second, then the preview playback
will also appear jerky, at 5 frames per second. A slider lets you drag the
animation to any frame from start to end.

Finally there are a couple of other tick boxes:

Hide messages during render. Hide all messages that may otherwise
appear in the rendered image/video.

Open after exporting. If you tick this box, then the image/video
will be opened after it is saved. It is opened in the same way as it would be
if you double-clicked on the file in Windows Explorer, so it will only work if
you have an application associated with the type of image/video file being
saved.

This is usually the first thing people ask me, and some people ask "Is the
fourth dimension time?". The world we experience every day has only three
spacial dimensions. Our brains have evolved to have a very intuitive
understanding of them, and it seems very "natural" to us that there must be
three dimensions, but it's really an arbitrary number and mathematically
there's no reason to stop at three. However, our brains have a great deal of
trouble grasping any more than three! Maybe this is where Stella4D can
help.

So the answer is that there is no fourth spacial dimension in our world, but
there are still ways to appreciate it. Mathematically, it just requires the
addition of one more mutually orthogonal axis, and indeed using cross-section
animations, you can use time as the fourth dimension!
See below.

When thinking about a four dimensional object, it is often beneficial to use an
analogy with 3D or 2D. By seeing how something works in 2D and 3D, it is often
easier to see how it should behave in 4D.

For example, each side of a polygon in 2D is an edge (a 1D line segment). Each
side of a polyhedron in 3D is a face (a 2D polygon). So by extension, each
side of a 4D polytope is a 3D polyhedron (which we call a cell). It is
still hard though to imagine a four dimensional region being bounded by
polyhedra.

Rotation is a tricky one. In 3D we require at least 3 values to specify a
rotation (heading, pitch, and roll). It is tempting then to think that 4D will
require 4 values, but this isn't the case. Consider 2D. Here only one value
is required to specify a rotation! So starting at 2D, we have 1, 3, then what?
Turns out these proceed as triangular numbers, i.e. 1, 3, 6, 10, 15, ...

The reason relates to how rotation works. In 2D a rotation happens
around a point (0D), in 3D a rotation happens around a line (1D), and by
extension in 4D a rotation happens around a plane (2D). Again, it's very hard
for us to imagine a rotation happening "around" a plane. A better way to think
of rotation is to place the emphasis on the coordinates that change, rather
than the ones that don't. When we say "rotation around a line", we mean that
the points along that line are not moved by the rotation. Instead we could
think about a plane in which the rotation takes place. For any-dimensional
space, rotation always affects coordinates in only two dimensions. So in 2D
there is only one way to choose those two dimensions, and in 3D there are three
ways (XY, XZ or YZ). In 4D we have six ways to choose two axes (XY, XZ, XW,
YZ, YW or ZW).

Although we can't experience a four-dimensional object directly in its original
form, there are several ways we can get a glimpse of them in 3D. The following
methods are supported in Stella4D:

Projection: A 4D object may be projected onto a 3D subspace, with
either an orthogonal or perspective projection, and then viewed in the same
manor as any 3D object. This is much like how 3D objects are viewed on 2D
computer screens. See 4D Projection.

Cross-Sections: Just as a slice through a 3D polyhedron gives a 2D
cross-section, a slice through a 4D polytope gives a 3D cross-section.
Changing the depth of the slicing plane in real-time creates amazing animations
which can be thought of as a higher dimensional polytope passing through a
lower dimensional space over time. Here, time really can be used as the fourth
dimension!
See Cross-Sections.

Nets: For 3D polyhedra, nets are 2D figures, consisting of a
collection of polygons (faces) connected together at edges. For 4D polytopes,
nets are 3D figures, consisting of polyhedra (cells) connected together at
faces. See 4D Nets.

Cells: the Cell View (same one that behaves as the Face View in 3D)
shows an individual cell of the polytope. Use its yellow left/right array
buttons to cycle through the different types of cell present.

Vertex Figures: the Vertex Figure view shows a 3D vertex figure for
the current vertex of the polytope. Use the yellow left/right arrow buttons
to cycle through the different types of vertex figure (note: uniform polytopes
only have one type of vertex figure). Each edge of the figure represents a
face of the polytope and is labelled with the number of sides that face has
(for a double-density polygon, a label like "5/2" will appear). Each vertex
represents an edge, and edge face represents a cell.

To get started in 4D, you will want to load one of the 4D polytopes.
From the category drop-down list in the main toolbar
select "4D Library". The model drop-down list to the right will then show a
list of sub-categories containing 4D models. Select one to switch to that
category. The list of sub-categories then moves to the category list (on the
left) and the model list shows the models available in the selected category
(on the right). The first model in that category will be loaded straight away,
and you can select other models from that category in the usual way (from the
drop-down list, with the green left/right buttons, or with the left/right arrow
keys).

Categories 1 through 29 are those defined by Jonathan Bowers and contain all
1849 known uniform 4D polytopes (the set is generally thought to be complete,
but not proven so). Note: a few models are repeated in order to group all
members of a regiment together, even when some members officially belong to
another category. Some additional categories appear after these:

CatA_Duoprisms. An assortment of duoprisms. Note: any duoprism can
be made using "4D→Create Duoprism...". Duoprisms are an infinite
series of uniform polychora.

CatB_Antiduoprisms. An assortment of antiduoprisms. Note: any
antiduoprism can be made using "4D→Create Antiduoprism...".
An antiduoprism is simply a 4D prism built on a 3D antiprism. This is another
infinite series of uniform polychora.

CatS1 to CatS9. Scaliform polychora, grouped into the categories
defined by Jonathan Bowers. These contain all known scaliforms, except for
categories S8 and S9, which should each contain 310 polytopes.

Compounds. A few interesting uniform compounds.

Convex. All the convex uniform 4D polytopes. These all appear
somewhere in categories 1 to 29, but this category groups them together for
convenience.

Fissary. These polytopes are uniform, but are not included in the
official count because they have either compound vertex figures or edge
figures.

Scaliform. Scaliform polytopes are vertex-uniform, and their faces
are regular, but not all of their cells are uniform.

The 4D polytopes all reside in the Stella4DLib folder under the
folder where Stella4D is installed. Uniform polytopes are stored not as
complete models, but only as their 3D vertex figures (this saves a lot of
disk-space). When loading, the full polytope is automatically generated (this
can be disabled with "4D→Auto-Generate Polychora from Vertex Figure
Files"). Normally when a file name starts with "verf-of-", an attempt is
made to use it as a vertex figure and generate a 4D model. You'll notice some
file names include something like "(3)" or "(5/2)" at the end. This is
required for cases where the choice of faces to be represented by each edge of
the vertex figure is ambiguous. The number in brackets represents the face
that should match the shortest edge of the vertex figure. Otherwise, for
ambiguous cases, a list of possibilities is presented, from which the user may
choose the appropriate option.

Not all the features from 3D are extended for use in 4D. You will notice that
many of the menus are greyed out when a 4D model is loaded. Some of the menu
labels change for 4D too, the most common being the change from "face" to
"cell".

Models from any supported view type can be used as the new base model.
The left-and-down arrow button appears at the top of each view, just like for
3D models, but a small "3" also appears on the button, indicating that clicking
it will create a 3D model. Use this if you want to create 2D nets for a paper
model of a cross-section, for example. The 4D views (dual, base + dual) also
have a similar button with a small "4" on it. Using that button creates a 4D
model. For example, the base + dual compound would become the new base model,
allowing you to see cross-sections of it etc.

Display of vertices and edges is supported, including
the spheres and cylinders option. For a perspective projection from 4D to 3D,
spheres and cylinders appear in different sizes depending on how far they were
from the 4D viewer.

Most coloring and hiding options for faces in 3D are supported for cells in
4D.

The symmetry group of a 4D polytope is not established. Instead,
cells/faces/edges/vertices are grouped into types by observing various
heuristics. It may sometimes group too many entities into a single type.
4D compounds are one example when this is likely to happen. This generally
isn't too important however.

Info window. Open the cells/faces/edges/vertices section for the
base or dual in the Info window and click on one of the
cell/face/edge/vertex types to select a cell/face/edge/vertex of that type.
You will see it highlighted in the main view. Note also that if the
appropriate section is open, then the appropriate entry in the Info window will
also be highlighted if you select a cell/face/edge/vertex in some other way.

Model views. In the base, dual, or base + dual views, you may
select items as follows:

Shift+Left-click: select a cell, edge or vertex

Shift+Right-click: select a vertex

Ctrl+Left-click: select a face

Ctrl+Right-click: select an edge

The last two are not available when the cross-section is also being displayed
in this view (based on the "Section→Show Cross-Section in Model
Views" submenu). In this case these are used to control the slice depth.

Cell view. In the individual cell view, you may select items as
follows:

Shift+Left-click: select a face, edge or vertex

Shift+Right-click: select a vertex

Ctrl+Left-click: select a face

Ctrl+Right-click: click on a face to select the neighbouring
cell at that face.

Vertex Figure view. In the vertex figure view, you may select items
as follows:

Shift+Left-click: click on a face to select the cell it
represents, an edge to select the face it represents, or a vertex to select
the edge it represents.

Shift+Right-click: click on a vertex to select the edge it
represents.

Ctrl+Left-click: click on a face to select the cell it
represents.

Ctrl+Right-click: click on a vertex to select the neighbouring
vertex along the edge it represents.

Cross-Section view. In the cross-section view, you may select items
as follows:

Shift+Left-click: click on a face to select the cell it
represents, an edge to select the face it represents, or a vertex to select
the edge it represents.

Shift+Right-click: click on a vertex to select the edge it
represents.

Unfolded Net views. In the unfolded net view, you may select items
as follows:

Shift+Left-click: select a cell, edge or vertex.

Shift+Right-click: select a vertex. All points in the net
that would fold together to the same 4D vertex are highlighted.

Ctrl+Left-click: select a face. The two faces that will fold
together are both highlighted.

By index. You can select a cell/face/edge/vertex by index using
"Selection→Select Cell by Index..." or "Selection→Select
Vertex by Index...". Both of these open the same window, but with a
different option selected. The window gives you the option of selecting a face
or edge by index too.

You don't have to remember all these combinations of Shift/Ctrl and Left/Right
mouse clicks though. Just remember to use the mouse tips
in the bottom right corner of the window.Tip: A double-click may be used instead of holding down Shift.
E.g. double left-click is the same as Shift+Left-click.

The base, dual, and base + dual views show a 4D polytope projected into 3D.
The projection may be either orthogonal or perspective, and front-facing or
back-facing cells may be hidden from the view (see The 4D
Menu).

In the default mode, the following operations are available with the mouse.

Shift+Left-click: select a cell, edge or vertex.

Shift+Right-click: select a vertex.

Ctrl+Left-click: select a face, or if the cross-section is
also being displayed embedded in this view, then drag to change the sectioning
depth.

Ctrl+Right-click: select an edge, or if the cross-section
is also being displayed embedded in this view, then click to set the sectioning
depth to 0.5.

Ctrl+Shift+Left-drag: Scale the displayed size of cells.
This shrinks the cells, but keeps their edges (if displayed) at full size. It
often helps with visualization to be able to see between cells in this way.

Ctrl+Shift+Right-click: Unscale the cells. Unless you hide
some cells, it will generally be difficult to see what's going on with unscaled
(full-size) cells, since they overlap and much detail is hidden.

The normal mouse navigation controls continue to
work, allowing you to rotate the displayed 3D projection in the usual ways.
But if you want to actually rotate the model in 4D, before being projected into
3D, then you need to enter 4D Tumble Mode
("Selection→Mouse Selection Modes→4D Tumble Mode" or the
matching toolbar button). Here the following mouse operations are available.

Shift+Left-drag: 4D rotate.

Shift+Right-drag: 4D tunnel. Rotates out towards you.

Ctrl+Left-drag: 4D rotate through cell. Rotates through
a particular cell. If you start the mouse-drag by clicking on a cell, then
that cell is used. If you click on the background, then the most recently
selected cell is used.

Ctrl+Right-drag: 4D rotate through vertex. Similar to
above but using vertices.

Space+Right-drag: Change field-of-view used for 4D
perspective projection. The viewer is also moved towards or away from the
model to compensate, keeping the model about the same size on the screen so
that only the perspective appears to change. At one extreme the projection
will look almost orthogonal, at the other it can produce a Schlegel
diagram, where the viewer is so close to a cell that all other cells are
hidden behind it.

You probably have to try each of the above to really get a feel for them.
Mouse inertia is supported for all of these, so if you release the mouse
button while still moving the mouse, the rotation will continue on its own.

The first two operations above are view-dependent. That is, the direction
of rotation relative to the model depends on which direction you are looking at
it from. This makes it intuitive regardless of your viewing direction. The
second two operations are model-dependent, and behave the same regardless of
your viewing direction. The difference is particularly noticeable if you
rotate the projected model in 3D while it continues rotating in 4D.

You can also choose from some specific 4D orientations of interest by using
the "4D→Projection Direction" submenu.

In 4D, nets are 3D figures, consisting of polyhedra (cells) connected together
at faces. Use the Unfolded Net view to see these for any loaded 4D polytope
("View→View Nets→Unfolded Net"). Stella4D attempts to
generate nets with as much symmetry and aesthetic appeal as possible.

One may ponder the usefulness of 3D nets for 4D polytopes, given that we
could never fold them up in our world to form the model of interest, but they
are still a useful way to see how the cells of a polytope fit together, and
have a beauty all of their own.

Note: in 3D nets are generated using only the externally visible parts of
faces. In 4D, since the nets can never be folded up anyway, intersections
between cells are ignored, and complete intact cells are used in the nets.
Nets are of most use in understanding convex polytopes, so this difference is
not too important.

Various interesting operations are available in the Unfolded Net view:

Shift+Left-click: select a cell, edge or vertex.

Shift+Right-click: select a vertex. All points in the net
that would fold together to the same 4D vertex are highlighted.

Ctrl+Left-click: select a face. The matching face
elsewhere in the net is also highlighted. This can help you to understand how
the net would fold together.

Ctrl+Right-click: rearrange the net by gluing the
neighbouring cell to the face being clicked on (removing it from wherever it
was before). Other cells attached to the neighbouring cell will also have to
come along for the ride if they would otherwise no longer be part of the net.

Ctrl+Shift+Left-drag: Scale the displayed size of cells in
the net. This shrinks the cells, but keeps their edges (if displayed) at full
size. It sometimes helps with visualization to be able to see between cells in
this way.

Ctrl+Shift+Right-click: Unshrink the cells.

Manually rearranged nets are remembered in the .stel file. If you
want to build up the net one cell at a time, rather than rearranging an already
complete net, you can use "Nets→Maximum Cells per Net". Set that
value to "1" to force every cell into a separate net. Then you may use
Ctrl+Right-click to grow the net one cell at a time.

Perspective. Use a perspective projection (cells that are
further away will appear smaller).

Orthogonal. Use an orthogonal projection.

Same as 3D Setting (the default). Use the same type of
projection for both 4D-to-3D and 3D-to-2D. See
Navigation for a discussion of perspective and
orthogonal projections in 3D.

Projection Direction submenu. For each option below, the
selected (or most recently selected) entity is used.Tip: Right-click on the 4D Tumble Mode toolbar button to open a menu
with these same choices.

Cell First. The 4D viewer is looking directly at a cell.

Face First. The 4D viewer is looking directly at a face.

Edge First. The 4D viewer is looking directly at an edge.

Vertex First. The 4D viewer is looking directly at a vertex.

Measured Item First. The 4D viewer is looking directly at the
most recent item defined in Measurement Mode.

Automatic. Use whichever of the above gives the greatest
symmetry.

You may also use 4D Tumble mode (see 4D Projection) to
interactively change the projection direction.

Hide Back-Cells. When performing the projection from 4D to 3D, leave
out cells which are facing away from the 4D viewer. This makes most sense for
convex polytopes, and is similar to the case in 3D, where faces behind a
polyhedron are not visible to the viewer.

Hide Front-Cells(ticked by default). When performing the
projection from 4D to 3D, leave out cells which are facing towards the 4D
viewer. This makes most sense for convex polytopes. It is similar to what
would happen in 3D if the visible faces were removed from a convex polyhedron.
You would see inside instead. This works well when a strong perspective
projection is used, which is like looking at a polyhedron with your eye very
close to one face. If close enough, that face may be the only one you can see,
with all others hidden behind it. Hiding the face in front allows all other
faces to be seen at once (a Schlegel diagram).

When Hiding Cells, Hide Vertices/Edges Too. With this enabled,
vertices and edges that aren't part of any visible cell are also hidden. Only
applies to views showing a 4D model projected into 3D. Can be useful for
avoiding a huge tangle of edges when studying just a few cells.

Truncate. Truncate the current polytope. Each vertex becomes a new
cell, old cells are truncated.

Rectify. Rectify the current polytope. This is like truncating
deeper, until the edges between original vertices shrink to nothing. It is
only possible when the polytope has all edges tangent to a common sphere (the
tangency points will become new vertices).

Expand/Runcinate. Expand the current polytope. The existing cells
are spread out, with the dual cells inserted between them, and prisms appear to
fill up the gaps.

Create Duoprism.... Create an arbitrary duoprism based on any two
regular polygons. A prompt will appear asking for a definition of the two
polygons to be used. Each takes the form "n" or "n/d" where n and
d are positive integers. For example, "5" is a pentagon or "5/2" is a
pentagram. The two values should be separated by a space. A single value may
be entered if you wish both polygons to be the same.

Create Antiduoprism.... Create an arbitrary antiduoprism based on
any regular polygon. A prompt will appear asking for a definition of the
polygon to be used. The format is the same as for duoprisms above. An
antiduoprism is simply a 4D prism built upon a 3D antiprism. Note: for crossed
antiprisms, use a retrograde polygon like "5/3".

Create Step Prism (Duals of Gyrochora).... Allows you to enter two
numbers which define a step prism. The first represents a polygon, and
the second the step size. The duals of these are the gyrochora which
are an interesting set of 4D dice, having all cells identical, but a weird
batch of creatures they are, mostly with highly irregular-looking cells!

Create 4D Prism on Current Polyhedron. Build a 4D prism using the
current polyhedron as a base.

Create Uniform Polychoron from 3D Vertex Figure(keyboard
shortcut: U). Attempt to create a uniform (or scaliform) 4D polytope using
the current polyhedron as a vertex figure. If regular faces can be fitted to
the vertex figure's edges in multiple ways, a number of choices is presented to
choose from. You will need to know what you're doing though to find polyhedra
that work as vertex figures. All known uniform polychora are already provided
with Stella4D, but other scaliform, coincidic and fissary cases still
exist.

Auto-Generate Polychora from Vertex Figure Files. The 4D polytopes
provided with Stella4D are mostly just saved as vertex figures (which
saves a huge amount of disk space). When loaded, the 4D model they represent
is automatically generated. You can disable this behaviour however by
unticking this menu item. Then you can load the vertex figure models
themselves. A vertex figure file is recognised by a file name that starts with
"verf-of-". See also Loading 4D Polytopes for further
discussion of vertex figure file names.

Hopefully you'll be able to work the rest of the interface out on your own!
There are lots of menus to look through. Let me know if you have any problems
or can't figure out how to do something. I'll be interested to hear what you
think!